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Register a new userDeriving Potential Coronal Magnetic Fields from Vector Magnetograms
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The minimum-energy configuration for the magnetic field above the solar photosphere is curl-free (hence, by Amperes law, also current-free), so can be represented as the gradient of a scalar potential. Since magnetic fields are divergence free, this scalar potential obeys Laplaces equation, given an appropriate boundary condition (BC). With measurements of the full magnetic vector at the photosphere, it is possible to employ either Neumann or Dirichlet BCs there. Historically, the Neumann BC was used with available line-of-sight magnetic field measurements, which approximate the radial field needed for the Neumann BC. Since each BC fully determines the 3D vector magnetic field, either choice will, in general, be inconsistent with some aspect of the observed field on the boundary, due to the presence of both currents and noise in the observed field. We present a method to combine solutions from both Dirichlet and Neumann BCs to determine a hybrid, least-squares potential field, which minimizes the integrated square of the residual between the potential and actual fields. This has advantages in both not overfitting the radial field used for the Neumann BC, and maximizing consistency with the observations. We demonstrate our methods with SDO/HMI vector magnetic field observations of AR 11158, and find that residual discrepancies between the observed and potential fields are significant, and are consistent with nonzero horizontal photospheric currents. We also analyze potential fields for two other active regions observed with two different vector magnetographs, and find that hybrid potential fields have significantly less energy than the Neumann fields in every case --- by more than 10^(32) erg in some cases. This has major implications for estimates of free magnetic energy in coronal field models, e.g., non-linear force-free field extrapolations.
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The SDO/HMI instruments provide photospheric vector magnetograms with a high spatial and temporal resolution. Our intention is to model the coronal magnetic field above active regions with the help of a nonlinear force-free extrapolation code. Our code is based on an optimization principle and has been tested extensively with semi-analytic and numeric equilibria and been applied before to vector magnetograms from Hinode and ground based observations. Recently we implemented a new version which takes measurement errors in photospheric vector magnetograms into account. Photospheric field measurements are often due to measurement errors and finite nonmagnetic forces inconsistent as a boundary for a force-free field in the corona. In order to deal with these uncertainties, we developed two improvements: 1.) Preprocessing of the surface measurements in order to make them compatible with a force-free field 2.) The new code keeps a balance between the force-free constraint and deviation from the photospheric field measurements. Both methods contain free parameters, which have to be optimized for use with data from SDO/HMI. Within this work we describe the corresponding analysis method and evaluate the force-free equilibria by means of how well force-freeness and solenoidal conditions are fulfilled, the angle between magnetic field and electric current and by comparing projections of magnetic field lines with coronal images from SDO/AIA. We also compute the available free magnetic energy and discuss the potential influence of control parameters.
Context: Knowledge about the coronal magnetic field is important to the understanding the structure of the solar corona. We compute the field in the higher layers of the solar atmosphere from the measured photospheric field under the assumption that the corona is force-free. Aims: Here we develop a method for nonlinear force-free coronal magnetic field medelling and preprocessing of photospheric vector magnetograms in spherical geometry using the optimization procedure. Methods: We describe a newly developed code for the extrapolation of nonlinear force-free coronal magnetic fields in spherical coordinates over a restricted area of the Sun. The program uses measured vector magnetograms on the solar photosphere as input and solves the force-free equations in the solar corona. We develop a preprocessing procedure in spherical geometry to drive the observed non-force-free data towards suitable boundary conditions for a force-free extrapolation. Results: We test the code with the help of a semi-analytic solution and assess the quality of our reconstruction qualitatively by magnetic field line plots and quantitatively with a number of comparison metrics for different boundary conditions. The reconstructed fields from the lower boundary data with the weighting function are in good agreement with the original reference fields. We added artificial noise to the boundary conditions and tested the code with and without preprocessing. The preprocessing recovered all main structures of the magnetogram and removed small-scale noise. The main test was to extrapolate from the noisy photospheric vector magnetogram with and without preprocessing. The preprocessing was found to significantly improve the agreement between the extrapolated and the exact field.
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.
The availability of vector magnetogram sequences with sufficient accuracy and cadence to estimate the time derivative of the magnetic field allows us to use Faradays law to find an approximate solution for the electric field in the photosphere, using a Poloidal-Toroidal Decomposition (PTD) of the magnetic field and its partial time derivative. Without additional information, however, the electric field found from this technique is under-determined -- Faradays law provides no information about the electric field that can be derived the gradient of a scalar potential. Here, we show how additional information in the form of line-of-sight Doppler flow measurements, and motions transverse to the line-of-sight determined with ad-hoc methods such as local correlation tracking, can be combined with the PTD solutions to provide much more accurate solutions for the solar electric field, and therefore the Poynting flux of electromagnetic energy in the solar photosphere. Reliable, accurate maps of the Poynting flux are essential for quantitative studies of the buildup of magnetic energy before flares and coronal mass ejections.
We estimated the accuracy of coronal magnetic fields derived from radio observations by comparing them to potential field calculations and the DEM measurements using EUV observations. We derived line of sight component of the coronal magnetic field from polarization observations of the thermal bremsstrahlung in the NOAA active region 11150, observed around 3:00 UT on February 3, 2011 using the Nobeyama Radioheliograph at 17 GHz. Because the thermal bremsstrahlung intensity at 17 GHz includes both chromospheric and coronal components, we extracted only the coronal component by measuring the coronal emission measure in EUV observations. In addition, we derived only the radio polarization component of the corona by selecting the region of coronal loops and weak magnetic field strength in the chromosphere along the line of sight. The upper limit of the coronal longitudinal magnetic fields were determined as 100 - 210 G. We also calculated the coronal longitudinal magnetic fields from the potential field extrapolation using the photospheric magnetic field obtained from the Helioseismic and Magnetic Imager (HMI). However, the calculated potential fields were certainly smaller than the observed coronal longitudinal magnetic field. This discrepancy between the potential and the observed magnetic field strengths can be explained consistently by two reasons; (a) the underestimation of the coronal emission measure resulting from the limitation of the temperature range of the EUV observations, (b) the underestimation of the coronal magnetic field resulting from the potential field assumption.
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