In the quiet solar photosphere, the mixed polarity fields form a magnetic carpet, which continuously evolves due to dynamical interaction between the convective motions and magnetic field. This interplay is a viable source to heat the solar atmospher
e. In this work, we used the line-of-sight (LOS) magnetograms obtained from the Helioseismic and Magnetic Imager (HMI) on the textit{Solar Dynamics Observatory} (textit{SDO}), and the Imaging Magnetograph eXperiment (IMaX) instrument on the textit{Sunrise} balloon-borne observatory, as time dependent lower boundary conditions, to study the evolution of the coronal magnetic field. We use a magneto-frictional relaxation method, including hyperdiffusion, to produce time series of three-dimensional (3D) nonlinear force-free fields from a sequence of photospheric LOS magnetograms. Vertical flows are added up to a height of 0.7 Mm in the modeling to simulate the non-force-freeness at the photosphere-chromosphere layers. Among the derived quantities, we study the spatial and temporal variations of the energy dissipation rate, and energy flux. Our results show that the energy deposited in the solar atmosphere is concentrated within 2 Mm of the photosphere and there is not sufficient energy flux at the base of the corona to cover radiative and conductive losses. Possible reasons and implications are discussed. Better observational constraints of the magnetic field in the chromosphere are crucial to understand the role of the magnetic carpet in coronal heating.
The solar atmosphere may be heated by Alfven waves that propagate up from the convection zone and dissipate their energy in the chromosphere and corona. To further test this theory, we consider wave heating in an active region observed on 2012 March
7. A potential field model of the region is constructed, and 22 field lines representing observed coronal loops are traced through the model. Using a three-dimensional (3D) reduced magneto-hydrodynamics (MHD) code, we simulate the dynamics of Alfven waves in and near the observed loops. The results for different loops are combined into a single formula describing the average heating rate Q as function of position within the observed active region. We suggest this expression may be approximately valid also for other active regions, and therefore may be used to construct 3D, time-dependent models of the coronal plasma. Such models are needed to understand the role of thermal non-equilibrium in the structuring and dynamics of the Suns corona.
The structure of the coronal magnetic field prior to eruptive processes and the conditions for the onset of eruption are important issues that can be addressed through studying the magnetohydrodynamic stability and evolution of nonlinear force-free f
ield (NLFFF) models. This paper uses data-constrained NLFFF models of a solar active region that erupted on 2010 April 8 as initial condition in MHD simulations. These models, constructed with the techniques of flux rope insertion and magnetofrictional relaxation, include a stable, an approximately marginally stable, and an unstable configuration. The simulations confirm previous related results of magnetofrictional relaxation runs, in particular that stable flux rope equilibria represent key features of the observed pre-eruption coronal structure very well and that there is a limiting value of the axial flux in the rope for the existence of stable NLFFF equilibria. The specific limiting value is located within a tighter range, due to the sharper discrimination between stability and instability by the MHD description. The MHD treatment of the eruptive configuration yields very good agreement with a number of observed features like the strongly inclined initial rise path and the close temporal association between the coronal mass ejection and the onset of flare reconnection. Minor differences occur in the velocity of flare ribbon expansion and in the further evolution of the inclination; these can be eliminated through refined simulations. We suggest that the slingshot effect of horizontally bent flux in the source region of eruptions can contribute significantly to the inclination of the rise direction. Finally, we demonstrate that the onset criterion formulated in terms of a threshold value for the axial flux in the rope corresponds very well to the threshold of the torus instability in the considered active region.
We used data from the Helioseismic and Magnetic Imager (HMI), and Atmospheric Imaging Assembly (AIA) on the textit{Solar Dynamics Observatory} (SDO) to study coronal loops at small scales, emerging in the quiet Sun. With HMI line-of-sight magnetogram
s, we derive the integrated and unsigned photospheric magnetic flux at the loop footpoints in the photosphere. These loops are bright in the EUV channels of AIA. Using the six AIA EUV filters, we construct the differential emission measure (DEM) in the temperature range $5.7 - 6.5$ in log $T$ (K) for several hours of observations. The observed DEMs have a peak distribution around log $T approx$ 6.3, falling rapidly at higher temperatures. For log $T <$ 6.3, DEMs are comparable to their peak values within an order of magnitude. The emission weighted temperature is calculated, and its time variations are compared with those of magnetic flux. We present two possibilities for explaining the observed DEMs and temperatures variations. (a) Assuming the observed loops are comprised of hundred thin strands with certain radius and length, we tested three time-dependent heating models and compared the resulting DEMs and temperatures with the observed quantities. This modeling used Enthalpy-based Thermal Evolution of Loops (EBTEL), a zero-dimensional (0D) hydrodynamic code. The comparisons suggest that a medium frequency heating model with a population of different heating amplitudes can roughly reproduce the observations. (b) We also consider a loop model with steady heating and non-uniform cross-section of the loop along its length, and find that this model can also reproduce the observed DEMs, provided the loop expansion factor $gamma sim$ 5 - 10. More observational constraints are required to better understand the nature of coronal heating in the short emerging loops on the quiet Sun.
This paper is the third in a series of papers working towards the construction of a realistic, evolving, non-linear force-free coronal field model for the solar magnetic carpet. Here, we present preliminary results of 3D time-dependent simulations of
the small-scale coronal field of the magnetic carpet. Four simulations are considered, each with the same evolving photospheric boundary condition: a 48 hr time series of synthetic magnetograms produced from the model of Meyer, Mackay, van Ballegooijen and Parnell, 2011, Solar Phys., 272, 29. Three simulations include a uniform, overlying coronal magnetic field of differing strength, the fourth simulation includes no overlying field. The build-up, storage and dissipation of magnetic energy within the simulations is studied. In particular, we study their dependence upon the evolution of the photospheric magnetic field and the strength of the overlying coronal field. We also consider where energy is stored and dissipated within the coronal field. The free magnetic energy built up is found to be more than sufficient to power small-scale, transient phenomena such as nanoflares and X-ray bright points, with the bulk of the free energy found to be stored low down, between 0.5-0.8 Mm. The energy dissipated is presently found to be too small to account for the heating of the entire quiet Sun corona. However, the form and location of energy dissipation regions are in qualitative agreement with what is observed on small scales on the Sun. Future MHD modelling using the same synthetic magnetograms may lead to a higher energy release.
This paper is the second in a series of studies working towards constructing a realistic, evolving, non-potential coronal model for the solar magnetic carpet. In the present study, the interaction of two magnetic elements is considered. Our objective
s are to study magnetic energy build up, storage and dissipation as a result of emergence, cancellation, and flyby of these magnetic elements. In the future these interactions will be the basic building blocks of more complicated simulations involving hundreds of elements. Each interaction is simulated in the presence of an overlying uniform magnetic field, which lies at various orientations with respect to the evolving magnetic elements. For these three small-scale interactions, the free energy stored in the field at the end of the simulation ranges from $0.2-2.1times 10^{26}$ ergs, while the total energy dissipated ranges from $1.3-6.3times 10^{26}$ ergs. For all cases, a stronger overlying field results in higher energy storage and dissipation. For the cancellation and emergence simulations, motion perpendicular to the overlying field results in the highest values. For the flyby simulations, motion parallel to the overlying field gives the highest values. In all cases, the free energy built up is sufficient to explain small-scale phenomena such as X-ray bright points or nanoflares. In addition, if scaled for the correct number of magnetic elements for the volume considered, the energy continually dissipated provides a significant fraction of the quiet Sun coronal heating budget.
The sub-arcsec bright points (BP) associated with the small scale magnetic fields in the lower solar atmosphere are advected by the evolution of the photospheric granules. We measure various quantities related to the horizontal motions of the BPs obs
erved in two wavelengths, including the velocity auto-correlation function. A 1 hr time sequence of wideband H$alpha$ observations conducted at the textit{Swedish 1-m Solar Telescope} (textit{SST}), and a 4 hr textit{Hinode} textit{G}-band time sequence observed with the Solar Optical telescope are used in this work. We follow 97 textit{SST} and 212 textit{Hinode} BPs with 3800 and 1950 individual velocity measurements respectively. For its high cadence of 5 s as compared to 30 s for textit{Hinode} data, we emphasize more on the results from textit{SST} data. The BP positional uncertainty achieved by textit{SST} is as low as 3 km. The position errors contribute 0.75 km$^2$ s$^{-2}$ to the variance of the observed velocities. The textit{raw} and textit{corrected} velocity measurements in both directions, i.e., $(v_x,v_y)$, have Gaussian distributions with standard deviations of $(1.32,1.22)$ and $(1.00, 0.86)$ km s$^{-1}$ respectively. The BP motions have correlation times of about $22 - 30$ s. We construct the power spectrum of the horizontal motions as a function of frequency, a quantity that is useful and relevant to the studies of generation of Alfven waves. Photospheric turbulent diffusion at time scales less than 200 s is found to satisfy a power law with an index of 1.59.
A three-dimensional MHD model for the propagation and dissipation of Alfven waves in a coronal loop is developed. The model includes the lower atmospheres at the two ends of the loop. The waves originate on small spatial scales (less than 100 km) ins
ide the kilogauss flux elements in the photosphere. The model describes the nonlinear interactions between Alfven waves using the reduced MHD approximation. The increase of Alfven speed with height in the chromosphere and transition region (TR) causes strong wave reflection, which leads to counter-propagating waves and turbulence in the photospheric and chromospheric parts of the flux tube. Part of the wave energy is transmitted through the TR and produces turbulence in the corona. We find that the hot coronal loops typically found in active regions can be explained in terms of Alfven wave turbulence, provided the small-scale footpoint motions have velocities of 1-2 km/s and time scales of 60-200 s. The heating rate per unit volume in the chromosphere is 2 to 3 orders of magnitude larger than that in the corona. We construct a series of models with different values of the model parameters, and find that the coronal heating rate increases with coronal field strength and decreases with loop length. We conclude that coronal loops and the underlying chromosphere may both be heated by Alfvenic turbulence.
