2D MHD modeling of magnetic reconnection in the stratified solar corona

The outermost layer of the solar atmosphere is the solar corona where the temperature exceeds one million degrees. The constituent gases are therefore highly ionized (plasma). Although it extends over several solar radii, the bright regions of the corona consist of closed almost semicircular magnetic tubes where the plasma is confined and heated, the coronal loops. An important role in the solar corona is played by magnetic reconnection. It is a fundamental phenomenon in electrically conducting fluids, in particular in plasmas. The basic picture underlying the idea of reconnection is that of two magnetic field lines (thin flux tubes, properly speaking) carried along with the fluid owing to the property of flux conservation, which come close together at some point, where by the effect of finite resistivity they are cut and reconnected. This process is in general expected to occur at small scales and releases magnetic energy in the form of thermal and kinetic energy and could therefore explain the heating of the corona. Recently, it has been proposed (Priest & Syntelis 2018) that opposite polarity magnetic flux tubes emerging from the solar surface are dragged around by solar photospheric motions and, coming into close contact, can disappear partially or totally, through magnetic reconnection (flux cancellation model), thus releasing large amounts of heating. An interesting question is then where this reconnection takes place along the flux tubes. In a uniform medium the reconnection site is essentially dictated by the symmetry of the system. However, the solar atmosphere and the coronal loops are highly non-uniform. We then approach the problem gradually, by investigating the reconnection site in a plasma stratified by the gravity, with a scale height on the order of several thousands of km, typical of coronal loops. Magnetic reconnection has been extensively investigated using MHD modeling (Chen et al. 1999, Yokoyama & Shibata 2001). Assuming an anomalous resistivity allows us to model reconnection in a domain with such large extension. In recent numerical MHD simulations by Ruan et al., 2020 and Leake et al., 2020, the reconnection site is forced, by assuming the anomalous resistivity only at a specific domain location. To focus on determining the reconnection site in a stratified coronal medium, we assume instead that the resistivity is the same everywhere in the medium. After defining a theoretical framework, describing the basics of magnetic reconnection and the important scaling laws, we model reconnection with 2 MHD simulations using a state-of-art numerical code, the PLUTO code (Mignone et al. 2007). As a reference case, we first model the reconnection between two magnetic slabs with opposite polarity separated by a current sheet in a uniform medium. We ascertain that the reconnection occurs symmetrically in a few Alfvén times (the time taken by a magnetic signal to travel across the plasma domain), with the formation of magnetic islands, whose size is in agreement with the scaling laws, and then the configuration remains steady. The core of this work is to replicate the MHD simulations for a stratified coronal medium. We define three important scaling lengths that determine the site of the reconnection, which is related to the size of the magnetic island, and in particular the length and the thickness of the initial current sheet, and the scale height determined by the pressure gradient that balances the gravity, which is ultimately connected to the ambient temperature. We performed a series of 2D MHD simulations with the PLUTO code, with different values of the 3 parameters and we find the following main results: the reconnection invariably occurs in the higher and less dense layers of the atmosphere, and in a few Alfvén times; the system becomes steady in about 20 Alfvén times; the thicker and shorter the initial current sheet, the lower is the reconnection site. We find instead an unexpected dependence on the pressure scale height. In particular, we find a bimodal behaviour: if the scale height is much shorter than the length of the current sheet, the height of the reconnection site moves lower as the scale height increases; if it is not, the reconnection site is invariably close to the top of the domain, i.e., it coincides with the length of the current sheet. As a final analysis, we have also checked the energy balance, and ascertained that the magnetic energy is released first into thermal and kinetic energy, and eventually exclusively into thermal energy when the system settles to a steady state, after about 5 Alfvén times. This work is a first step towards a more exhaustive and realistic modeling of reconnection in coronal loop