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The structure and dynamics of the solar corona is dominated by the magnetic field. In most areas in the corona magnetic forces are so dominant that all non-magnetic forces like plasma pressure gradient and gravity can be neglected in the lowest order. This model assumption is called the force-free field assumption, as the Lorentz force vanishes. This can be obtained by either vanishing electric currents (leading to potential fields) or the currents are co-aligned with the magnetic field lines. First we discuss a mathematically simpler approach that the magnetic field and currents are proportional with one global constant, the so-called linear force-free field approximation. In the generic case, however, the relation between magnetic fields and electric currents is nonlinear and analytic solutions have been only found for special cases, like 1D or 2D configurations. For constructing realistic nonlinear force-free coronal magnetic field models in 3D, sophisticated numerical computations are required and boundary conditions must be obtained from measurements of the magnetic field vector in the solar photosphere. This approach is currently of large interests, as accurate measurements of the photospheric field become available from ground-based (for example SOLIS) and space-born (for example Hinode and SDO) instruments. If we can obtain accurate force-free coronal magnetic field models we can calculate the free magnetic energy in the corona, a quantity which is important for the prediction of flares and coronal mass ejections. Knowledge of the 3D structure of magnetic field lines also help us to interpret other coronal observations, e.g., EUV-images of the radiating coronal plasma.
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Extrapolations of solar photospheric vector magnetograms into three-dimensional magnetic fields in the chromosphere and corona are usually done under the assumption that the fields are force-free. The field calculations can be improved by preprocessing the photospheric magnetograms. We compare two preprocessing methods presently in use, namely the methods of Wiegelmann et al. (2006) and Fuhrmann et al. (2007). The two preprocessing methods were applied to a recently observed vector magnetogram. We examine the changes in the magnetogram effected by the two preprocessing algorithms. Furthermore, the original magnetogram and the two preprocessed magnetograms were each used as input data for nonlinear force-free field extrapolations by means of two different methods, and we analyze the resulting fields. Both preprocessing methods managed to significantly decrease the magnetic forces and magnetic torques that act through the magnetogram area and that can cause incompatibilities with the assumption of force-freeness in the solution domain. Both methods also reduced the amount of small-scale irregularities in the observed photospheric field, which can sharply worsen the quality of the solutions. For the chosen parameter set, the Wiegelmann et al. method led to greater changes in strong-field areas, leaving weak-field areas mostly unchanged, and thus providing an approximation of the magnetic field vector in the chromosphere, while the Fuhrmann et al. method weakly changed the whole magnetogram, thereby better preserving patterns present in the original magnetogram. Both preprocessing methods raised the magnetic energy content of the extrapolated fields to values above the minimum energy, corresponding to the potential field. Also, the fields calculated from the preprocessed magnetograms fulfill the solenoidal condition better than those calculated without preprocessing.
Magnetic flux generated and intensified by the solar dynamo emerges into the solar atmosphere, forming active regions (ARs) including sunspots. Existing theories of flux emergence suggest that the magnetic flux can rise buoyantly through the convection zone but is trapped at the photosphere, while its further rising into the atmosphere resorts to the Parker buoyancy instability. To trigger such an instability, the Lorentz force in the photosphere needs to be as large as the gas pressure gradient to hold up an extra amount of mass against gravity. This naturally results in a strongly non-force-free photosphere, which is indeed shown in typical idealized numerical simulations of flux tube buoyancy from below the photosphere into the corona. Here we conduct a statistical study of the extents of normalized Lorentz forces and torques in the emerging photospheric magnetic field with a substantially large sample of SDO/HMI vector magnetograms. We found that the photospheric field has a rather small Lorentz force and torque on average, and thus is very close to a force-free state, which is not consistent with theories as well as idealized simulations of flux emergence. Furthermore, the small extents of forces and torques seem not to be influenced by the emerging ARs size, the emergence rate, or the non-potentiality of the field. This result puts an important constraint on future development of theories and simulations of flux emergence.
We use our semi-analytic solution of the nonlinear force-free field equation to construct three-dimensional magnetic fields that are applicable to the solar corona and study their statistical properties for estimating the degree of braiding exhibited by these fields. We present a new formula for calculating the winding number and compare it with the formula for the crossing number. The comparison is shown for a toy model of two helices and for realistic cases of nonlinear force-free fields; conceptually the formulae are nearly the same but the resulting distributions calculated for a given topology can be different. We also calculate linkages, which are useful topological quantities that are independent measures of the contribution of magnetic braiding to the total free energy and relative helicity of the field. Finally, we derive new analytical bounds for the free energy and relative helicity for the field configurations in terms of the linking number. These bounds will be of utility in estimating the braided energy available for nano-flares or for eruptions.
Self-organization properties of sustained magnetized plasma are applied to selected solar data to understand solar magnetic fields. Torsional oscillations are speed-up and slow-down bands of the azimuthal flow that correlate with the solar cycle, and they imply the existence of a symmetric solar dynamo with a measured polar flux of 3x10^14 Wb. It is shown that the solar dynamo is thin (~0.1 Mm gradient scale size) and powerful (~10^23 W). These properties are found from the amplitude of the torsional oscillations and the relationship of their velocity contours to solar magnetograms supports the result. The dynamo has enough power to heat the chromosphere and to power the corona and the solar wind. The dynamo also causes a rigid rotation of the heliosphere out to at least the corona and the relationship of the rotation of the corona to solar magnetograms supports this result as well. The thin solar dynamo sustains a thin stable minimum energy state that seems to be covering most of the solar surface just below the photosphere. The magnetic field lines of the minimum energy state should be parallel to the solar surface and rotate with distance from the surface with 2{pi} radians of rotation in ~1 Mm Resistive diffusion helps to push the magnetic fields to the surface and the global magnetic structure (GMS) seems to lose {pi} radians every 11 years, causing the observed 180 degree flipping of the solar magnetic field. The thin sheets of magnetized plasma in solar prominences may be the lost thin sheets of the GMS. For completeness, the formation of sunspots, CMEs and flares is discussed.
Adopting the thermal free-free emission mechanism, the coronal and chromospheric magnetic fields are derived from the polarization and spectral observations with the Nobeyama Radioheliograph at 1.76 cm. The solar active regions (AR) located near the disk center observed on January 8, 2015 (AR 12257) and December 4, 2016 (AR 12615) are used for the estimate of the chromospheric and coronal magnetic fields with the microwave radio observations. We compare solar radio maps of active regions for both intensity and circularly polarized component with the optical maps from observations with the Helioseismic and Magnetic Imager and the chromosphere and corona transition region images obtained with the Atmospheric Imaging Assembly instrument, on board the Solar Dynamic Observatory. We notice from the comparison between radio maps of both AR that the circular polarization degree in the AR 12257 is about 2 percent, but the AR 12615 has a higher existent value by 3 percent. Radio observations provide us for direct measurements of magnetic fields in the chromospheric and coronal layers. We estimate the coronal magnetic fields using the Atmospheric Imaging Assembly observations by adopting magnetic loops in the corona over some patches with weak photospheric magnetic fields. The coronal magnetic field derived from the Atmospheric Imaging Assembly data was from 90 to 240 Gauss. We also study the coronal magnetic fields based on the structure of the extrapolated field, where the result of the magnetic fields was in the range from 35 to 145 Gauss, showing that the difference in the coronal magnetic fields between both results is attributed to the assumption of the force-free approximation.
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