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Register a new userSparse reconstruction of electric fields from radial magnetic data
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Authors
A. R. Yeates
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Accurate estimates of the horizontal electric field on the Suns visible surface are important not only for estimating the Poynting flux of magnetic energy into the corona but also for driving time-dependent magnetohydrodynamic models of the corona. In this paper, a method is developed for estimating the horizontal electric field from a sequence of radial-component magnetic field maps. This problem of inverting Faradays law has no unique solution. Unfortunately, the simplest solution (a divergence-free electric field) is not realistically localized in regions of non-zero magnetic field, as would be expected from Ohms law. Our new method generates instead a localized solution, using a basis pursuit algorithm to find a sparse solution for the electric field. The method is shown to perform well on test cases where the input magnetic maps are flux balanced, in both Cartesian and spherical geometries. However, we show that if the input maps have a significant imbalance of flux - usually arising from data assimilation - then it is not possible to find a localized, realistic, electric field solution. This is the main obstacle to driving coronal models from time sequences of solar surface magnetic maps.
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Reconstructing complex networks from measurable data is a fundamental problem for understanding and controlling collective dynamics of complex networked systems. However, a significant challenge arises when we attempt to decode structural information hidden in limited amounts of data accompanied by noise and in the presence of inaccessible nodes. Here, we develop a general framework for robust reconstruction of complex networks from sparse and noisy data. Specifically, we decompose the task of reconstructing the whole network into recovering local structures centered at each node. Thus, the natural sparsity of complex networks ensures a conversion from the local structure reconstruction into a sparse signal reconstruction problem that can be addressed by using the lasso, a convex optimization method. We apply our method to evolutionary games, transportation and communication processes taking place in a variety of model and real complex networks, finding that universal high reconstruction accuracy can be achieved from sparse data in spite of noise in time series and missing data of partial nodes. Our approach opens new routes to the network reconstruction problem and has potential applications in a wide range of fields.
Photospheric electric fields, estimated from sequences of vector magnetic field and Doppler measurements, can be used to estimate the flux of magnetic energy (the Poynting flux) into the corona and as time-dependent boundary conditions for dynamic models of the coronal magnetic field. We have modified and extended an existing method to estimate photospheric electric fields that combines a poloidal-toroidal (PTD) decomposition of the evolving magnetic field vector with Doppler and horizontal plasma velocities. Our current, more comprehensive method, which we dub the {bf P}TD-{bf D}oppler-{bf F}LCT {bf I}deal (PDFI) technique, can now incorporate Doppler velocities from non-normal viewing angles. It uses the texttt{FISHPACK} software package to solve several two-dimensional Poisson equations, a faster and more robust approach than our previous implementations. Here, we describe systematic, quantitative tests of the accuracy and robustness of the PDFI technique using synthetic data from anelastic MHD (texttt{ANMHD}) simulations, which have been used in similar tests in the past. We find that the PDFI method has less than $1%$ error in the total Poynting flux and a $10%$ error in the helicity flux rate at a normal viewing angle $(theta=0$) and less than $25%$ and $10%$ errors respectively at large viewing angles ($theta<60^circ$). We compare our results with other inversion methods at zero viewing angle, and find that our methods estimates of the fluxes of magnetic energy and helicity are comparable to or more accurate than other methods. We also discuss the limitations of the PDFI method and its uncertainties.
Determining the electric field (E-field) distribution on the Suns photosphere is essential for quantitative studies of how energy flows from the Suns photosphere, through the corona, and into the heliosphere. This E-field also provides valuable input for data-driven models of the solar atmosphere and the Sun-Earth system. We show how Faradays Law can be used with observed vector magnetogram time series to estimate the photospheric E-field, an ill-posed inversion problem. Our method uses a poloidal-toroidal decomposition (PTD) of the time derivative of the vector magnetic field. The PTD solutions are not unique; the gradient of a scalar potential can be added to the PTD E-field without affecting consistency with Faradays Law. We present an iterative technique to determine a potential function consistent with ideal MHD evolution; but this E-field is also not a unique solution to Faradays Law. Finally, we explore a variational approach that minimizes an energy functional to determine a unique E-field, similar to Longcopes Minimum Energy Fit. The PTD technique, the iterative technique, and the variational technique are used to estimate E-fields from a pair of synthetic vector magnetograms taken from an MHD simulation; and these E-fields are compared with the simulations known electric fields. These three techniques are then applied to a pair of vector magnetograms of solar active region NOAA AR8210, to demonstrate the methods with real data.
We use daily full-disk vector magnetograms from Vector Spectromagnetograph (VSM) on Synoptic Optical Long-term Investigations of the Sun (SOLIS) system to synthesize the first Carrington maps of the photospheric vector magnetic field. We describe these maps and make a comparison of observed radial field with the radial field estimate from LOS magnetograms. Further, we employ these maps to study the hemispheric pattern of current helicity density, Hc, during the rising phase of the solar cycle 24. Longitudinal average over the 23 consecutive solar rotations shows a clear signature of the hemispheric helicity rule, i.e. Hc is predominantly negative in the North and positive in South. Although our data include the early phase of cycle 24, there appears no evidence for a possible (systematic) reversal of the hemispheric helicity rule at the beginning of cycle as predicted by some dynamo models. Further, we compute the hemispheric pattern in active region latitudes (-30 deg le theta le 30 deg) separately for weak (100< |B_r| <500 G)and strong (|B_r|>1000 G) radial magnetic fields. We find that while the current helicity of strong fields follows the well-known hemispheric rule (i.e., theta . Hc < 0), H_c of weak fields exhibits an inverse hemispheric behavior (i.e., theta . Hc > 0) albeit with large statistical scatter. We discuss two plausible scenarios to explain the opposite hemispheric trend of helicity in weak and strong field region.
The design of modern instruments does not only imply thorough studies of instrumental effects but also a good understanding of the scientific analysis planned for the data. We investigate the reliability of Milne-Eddington (ME)
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