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<jats:title>Abstract</jats:title> <jats:p>In many statistical solar flare studies, power laws are claimed and exponents derived by fitting a line to a log–log histogram. It is well known that this approach is statistically unstable, and very large statistics are needed to produce reliable exponent estimates. This may explain part of the observed divergence in power-law exponents in various studies. Moreover, the question is seldom addressed to what extent the data really do support power-law behavior. In this paper, we perform a comprehensive study of 6924 flares detected in <jats:italic>SDO</jats:italic>/AIA 9.4 nm images by the Solar Demon flare detection software between 2010 May 13 and 2018 March 16 and 9601 flares detected during the same period in <jats:italic>GOES</jats:italic>/XRS data by the LYRAFF flare detection software. We apply robust statistics to the <jats:italic>SDO</jats:italic>/AIA 9.4 nm peak intensity and the <jats:italic>GOES</jats:italic>/XRS raw peak flux, background-subtracted peak flux, and background-subtracted fluence and find clear indications that most background-corrected data are not well described by a power law and that all are better described by a lognormal distribution, while the raw <jats:italic>GOES</jats:italic>/XRS peak flux is best described by a power law. This may explain the success of power-law fits in flare studies using uncorrected data. The behavior of flare distributions has important implications for large-scale science questions such as coronal heating and the nature of solar flares. The apparent lognormal character of flare distributions in our data sets suggests that the assumed power-law nature of flares and its consequences need to be reexamined with great care.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>Current helicity, <jats:italic>H</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub>, and magnetic helicity, <jats:italic>H</jats:italic> <jats:sub> <jats:italic>m</jats:italic> </jats:sub>, are two main quantities used to characterize magnetic fields. For example, such quantities have been widely used to characterize solar active regions and their ejecta (magnetic clouds). It is commonly assumed that <jats:italic>H</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub> and <jats:italic>H</jats:italic> <jats:sub> <jats:italic>m</jats:italic> </jats:sub> have the same sign, but this has not been rigorously addressed beyond the simple case of linear force-free fields. We aim to answer whether <jats:italic>H</jats:italic> <jats:sub> <jats:italic>m</jats:italic> </jats:sub> <jats:italic>H</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub> ≥ 0 in general, and whether it is true over some useful set of magnetic fields. This question is addressed analytically and with numerical examples. The main focus is on cylindrically symmetric straight flux tubes, referred to as flux ropes (FRs), using the relative magnetic helicity with respect to a straight (untwisted) reference field. Counterexamples with <jats:italic>H</jats:italic> <jats:sub> <jats:italic>m</jats:italic> </jats:sub> <jats:italic>H</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub> &lt; 0 have been found for cylindrically symmetric FRs with finite plasma pressure, and for force-free cylindrically symmetric FRs in which the poloidal field component changes direction. Our main result is a proof that <jats:italic>H</jats:italic> <jats:sub> <jats:italic>m</jats:italic> </jats:sub> <jats:italic>H</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub> ≥ 0 is true for force-free cylindrically symmetric FRs where the toroidal field and poloidal field components are each of a single sign, and the poloidal component does not exceed the toroidal component. We conclude that the conjecture that current and magnetic helicities have the same sign is not true in general, but it is true for a set of FRs of importance to coronal and heliospheric physics.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>Large-scale convective flows called giant cells were once thought to transport the Sun’s luminosity in the solar convection zone, but recent observations have called their existence into question. In place of large-scale flows, some authors have suggested the solar luminosity may instead be transported by small droplets of rapidly falling, low-entropy fluid. This “entropy rain” could propagate as dense vortex rings, analogous to rising buoyant thermals in the Earth’s atmosphere. In this work, we develop an analytical theory describing the evolution of dense, negatively buoyant thermals. We verify the theory with 2D cylindrical and 3D Cartesian simulations of laminar, axisymmetric thermals in highly stratified atmospheres. Our results show that dense thermals fall in two categories: a <jats:italic>stalling</jats:italic> regime in which the droplets slow down and expand, and a <jats:italic>falling</jats:italic> regime in which the droplets accelerate and shrink as they propagate downwards. We estimate that solar downflows are in the falling regime and maintain their entropy perturbation against diffusion until they reach the base of the convection zone. This suggests that entropy rain may be an effective nonlocal mechanism for transporting the solar luminosity.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>Thermal nonequilibrium (TNE) is a fascinating situation that occurs in coronal magnetic flux tubes (loops) for which no solution to the steady-state fluid equations exists. The plasma is constantly evolving even though the heating that produces the hot temperatures does not. This is a promising explanation for isolated phenomena such as prominences, coronal rain, and long-period pulsating loops, but it may also have much broader relevance. As known for some time, TNE requires that the heating be both (quasi-)steady and concentrated at low coronal altitudes. Recent studies indicate that asymmetries are also important, with large enough asymmetries in the heating and/or cross-sectional area resulting in steady flow rather than TNE. Using reasonable approximations, we have derived two formulae for quantifying the conditions necessary for TNE. As a rough rule of thumb, the ratio of apex to footpoint heating rates must be less than about 0.1, and asymmetries must be less than about a factor of 3. The precise values are case-dependent. We have tested our formulae with 1D hydrodynamic loop simulations and find a very acceptable agreement. These results are important for developing physical insight about TNE and assessing how widespread it may be on the Sun.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>Supernovae from core collapse of massive stars drive shocks into the molecular clouds from which the stars formed. Such shocks affect future star formation from the molecular clouds, and the fast-moving, dense gas with compressed magnetic fields is associated with enhanced cosmic rays. This paper presents new theoretical modeling, using the Paris–Durham shock model, and new observations at high spectral resolution, using the Stratospheric Observatory for Infrared Astronomy, of the H<jats:sub>2</jats:sub> S(5) pure rotational line from molecular shocks in the supernova remnant IC 443. We generate MHD models for nonsteady-state shocks driven by the pressure of the IC 443 blast wave into gas of densities 10<jats:sup>3</jats:sup>–10<jats:sup>5</jats:sup> cm<jats:sup>−3</jats:sup>. We present the first detailed derivation of the shape of the velocity profile for emission from H<jats:sub>2</jats:sub> lines behind such shocks, taking into account the shock age, preshock density, and magnetic field. For preshock densities 10<jats:sup>3</jats:sup>–10<jats:sup>5</jats:sup> cm<jats:sup>−3</jats:sup>, the H<jats:sub>2</jats:sub> emission arises from layers that extend 0.01–0.0003 pc behind the shock, respectively. The predicted shifts of line centers, and the line widths, of the H<jats:sub>2</jats:sub> lines range from 20–2 and 30–4 km s<jats:sup>−1</jats:sup>, respectively. The a priori models are compared to the observed line profiles, showing that clumps C and G can be explained by shocks into gas with density 10<jats:sup>3</jats:sup> to 2 × 10<jats:sup>4</jats:sup> cm<jats:sup>−3</jats:sup> and strong magnetic fields. Two positions in clump B were observed. For clump B2 (a fainter region near clump B), the H<jats:sub>2</jats:sub> spectrum requires a J-type shock into moderate-density gas (∼10<jats:sup>2</jats:sup> cm<jats:sup>−3</jats:sup>) with the gas accelerated to 100 km s<jats:sup>−1</jats:sup> from its preshock location. Clump B1 requires both a magnetic-dominated C-type shock (like for clumps C and G) and a J-type shock (like for clump B2) to explain the highest observed velocities. The J-type shocks that produce high-velocity molecules may be locations where the magnetic field is nearly parallel to the shock velocity, which makes it impossible for a C-type shock (with ions and neutrals separated) to form.</jats:p>