The Influence of Spatial Resolution on Nonlinear Force-Free Modeling

The nonlinear force-free field (NLFFF) model is often used to describe the solar coronal magnetic field, however a series of earlier studies revealed difficulties in the numerical solution of the model in application to photospheric boundary data. We investigate the sensitivity of the modeling to the spatial resolution of the boundary data, by applying multiple codes that numerically solve the NLFFF model to a sequence of vector magnetogram data at different resolutions, prepared from a single Hinode/SOT-SP scan of NOAA Active Region 10978 on 2007 December 13. We analyze the resulting energies and relative magnetic helicities, employ a Helmholtz decomposition to characterize divergence errors, and quantify changes made by the codes to the vector magnetogram boundary data in order to be compatible with the force-free model. This study shows that NLFFF modeling results depend quantitatively on the spatial resolution of the input boundary data, and that using more highly resolved boundary data yields more self-consistent results. The free energies of the resulting solutions generally trend higher with increasing resolution, while relative magnetic helicity values vary significantly between resolutions for all methods. All methods require changing the horizontal components, and for some methods also the vertical components, of the vector magnetogram boundary field in excess of nominal uncertainties in the data. The solutions produced by the various methods are significantly different at each resolution level. We continue to recommend verifying agreement between the modeled field lines and corresponding coronal loop images before any NLFFF model is used in a scientific setting.Comment: Accepted to ApJ; comments/corrections to this article are welcome via e-mail, even after publicatio

Review on the Brownian Dynamics Simulation of Bead-Rod-Spring Models Encountered in Computational Rheology

Kinetic theory is a mathematical framework intended to relate directly the most relevant characteristics of the molecular structure to the rheological behavior of the bulk system. In other words, kinetic theory is a micro-to-macro approach for solving the flow of complex fluids that circumvents the use of closure relations and offers a better physical description of the phenomena involved in the flow processes. Cornerstone models in kinetic theory employ beads, rods and springs for mimicking the molecular structure of the complex fluid. The generalized bead-rod-spring chain includes the most basic models in kinetic theory: the freely jointed bead-spring chain and the freely-jointed bead-rod chain. Configuration of simple coarse-grained models can be represented by an equivalent Fokker-Planck (FP) diffusion equation, which describes the evolution of the configuration distribution function in the physical and configurational spaces. FP equation can be a complex mathematical object, given its multidimensionality, and solving it explicitly can become a difficult task. Even more, in some cases, obtaining an equivalent FP equation is not possible given the complexity of the coarse-grained molecular model. Brownian dynamics can be employed as an alternative extensive numerical method for approaching the configuration distribution function of a given kinetic-theory model that avoid obtaining and/or resolving explicitly an equivalent FP equation. The validity of this discrete approach is based on the mathematical equivalence between a continuous diffusion equation and a stochastic differential equation as demonstrated by Itô in the 1940s. This paper presents a review of the fundamental issues in the BD simulation of the linear viscoelastic behavior of bead-rod-spring coarse grained models in dilute solution. In the first part of this work, the BD numerical technique is introduced. An overview of the mathematical framework of the BD and a review of the scope of applications are presented. Subsequently, the links between the rheology of complex fluids, the kinetic theory and the BD technique are established at the light of the stochastic nature of the bead-rod-spring models. Finally, the pertinence of the present state-of-the-art review is explained in terms of the increasing interest for the stochastic micro-to-macro approaches for solving complex fluids problems. In the second part of this paper, a detailed description of the BD algorithm used for simulating a small-amplitude oscillatory deformation test is given. Dynamic properties are employed throughout this work to characterise the linear viscoelastic behavior of bead-rod-spring models in dilute solution. In the third and fourth part of this article, an extensive discussion about the main issues of a BD simulation in linear viscoelasticity of diluted suspensions is tackled at the light of the classical multi-bead-spring chain model and the multi-bead-rod chain model, respectively. Kinematic formulations, integration schemes and expressions to calculate the stress tensor are revised for several classical models: Rouse and Zimm theories in the case of multi-bead-spring chains, and Kramers chain and semi-flexible filaments in the case of multi-bead-rod chains. The implemented BD technique is, on the one hand, validated in front of the analytical or exact numerical solutions known of the equivalent FP equations for those classic kinetic theory models; and, on the other hand, is control-set thanks to the analysis of the main numerical issues involved in a BD simulation. Finally, the review paper is closed by some concluding remarks

Testing non-linear force-free coronal magnetic field extrapolations with the Titov-Demoulin equilibrium

CONTEXT: As the coronal magnetic field can usually not be measured directly, it has to be extrapolated from photospheric measurements into the corona. AIMS: We test the quality of a non-linear force-free coronal magnetic field extrapolation code with the help of a known analytical solution. METHODS: The non-linear force-free equations are numerically solved with the help of an optimization principle. The method minimizes an integral over the force-free and solenoidal condition. As boundary condition we use either the magnetic field components on all six sides of the computational box in Case I or only on the bottom boundary in Case II. We check the quality of the reconstruction by computing how well force-freeness and divergence-freeness are fulfilled and by comparing the numerical solution with the analytical solution. The comparison is done with magnetic field line plots and several quantitative measures, like the vector correlation, Cauchy Schwarz, normalized vector error, mean vector error and magnetic energy. RESULTS: For Case I the reconstructed magnetic field shows good agreement with the original magnetic field topology, whereas in Case II there are considerable deviations from the exact solution. This is corroborated by the quantitative measures, which are significantly better for Case I. CONCLUSIONS: Despite the strong nonlinearity of the considered force-free equilibrium, the optimization method of extrapolation is able to reconstruct it; however, the quality of reconstruction depends significantly on the consistency of the input data, which is given only if the known solution is provided also at the lateral and top boundaries, and on the presence or absence of flux concentrations near the boundaries of the magnetogram.Comment: 6 pages, 2 figures, Research Not

Site effects of the Roio basin, L’Aquila

During the microzonation studies of the April 6th, 2009 L’Aquila earthquake, we observed local seismic amplifications in the Roio area—a plane separated from L’Aquila city center by mount Luco. Six portable, digital instruments were deployed across the plain from 15 April to mid-May 2009. This array recorded 152 aftershocks. We analyzed the ground motion from these events to determine relative site amplification within the plain and on surrounding ridges. Horizontal over vertical spectral ratio on noise data (HVSRN),aftershock recordings (HVEQ) and standard spectral ratio (SSR) showed amplifications at 1.3 and 4.0Hz on quaternary deposits. Seismic amplifications in the frequency range of 4 and 6Hz were also observed on a carbonate ridge of Colle di Roio, on the northwestern border of the plateau. A small amplification was noticed near the top of mount Luco, another rocky site. Large discrepancies in the amplification levels between methods have been observed for these sites, but the HVSRN, HVEQ and SSR gave similar results at the stations located in the Roio plain. On the rocky sites, the SSR was more reliable than the HVSRN at estimating the transfer function of the site, even if the resonance frequency seemed to be well detected by the latter method.Published809–8234.1. Metodologie sismologiche per l'ingegneria sismicaJCR Journalreserve

Bayesian Centroid Estimation for Motif Discovery

Biological sequences may contain patterns that are signal important biomolecular functions; a classical example is regulation of gene expression by transcription factors that bind to specific patterns in genomic promoter regions. In motif discovery we are given a set of sequences that share a common motif and aim to identify not only the motif composition, but also the binding sites in each sequence of the set. We present a Bayesian model that is an extended version of the model adopted by the Gibbs motif sampler, and propose a new centroid estimator that arises from a refined and meaningful loss function for binding site inference. We discuss the main advantages of centroid estimation for motif discovery, including computational convenience, and how its principled derivation offers further insights about the posterior distribution of binding site configurations. We also illustrate, using simulated and real datasets, that the centroid estimator can differ from the maximum a posteriori estimator.Comment: 24 pages, 9 figure

Testing magnetofrictional extrapolation with the Titov-D\'emoulin model of solar active regions

We examine the nonlinear magnetofrictional extrapolation scheme using the solar active region model by Titov and D\'emoulin as test field. This model consists of an arched, line-tied current channel held in force-free equilibrium by the potential field of a bipolar flux distribution in the bottom boundary. A modified version, having a parabolic current density profile, is employed here. We find that the equilibrium is reconstructed with very high accuracy in a representative range of parameter space, using only the vector field in the bottom boundary as input. Structural features formed in the interface between the flux rope and the surrounding arcade-"hyperbolic flux tube" and "bald patch separatrix surface"-are reliably reproduced, as are the flux rope twist and the energy and helicity of the configuration. This demonstrates that force-free fields containing these basic structural elements of solar active regions can be obtained by extrapolation. The influence of the chosen initial condition on the accuracy of reconstruction is also addressed, confirming that the initial field that best matches the external potential field of the model quite naturally leads to the best reconstruction. Extrapolating the magnetogram of a Titov-D\'emoulin equilibrium in the unstable range of parameter space yields a sequence of two opposing evolutionary phases which clearly indicate the unstable nature of the configuration: a partial buildup of the flux rope with rising free energy is followed by destruction of the rope, losing most of the free energy.Comment: 14 pages, 10 figure

Chromospheric seismology above sunspot umbrae

The acoustic resonator is an important model for explaining the three-minute oscillations in the chromosphere above sunspot umbrae. The steep temperature gradients at the photosphere and transition region provide the cavity for the acoustic resonator, which allows waves to be both partially transmitted and partially reflected. In this paper, a new method of estimating the size and temperature profile of the chromospheric cavity above a sunspot umbra is developed. The magnetic field above umbrae is modelled numerically in 1.5D with slow magnetoacoustic wave trains travelling along magnetic fieldlines. Resonances are driven by applying the random noise of three different colours---white, pink and brown---as small velocity perturbations to the upper convection zone. Energy escapes the resonating cavity and generates wave trains moving into the corona. Line of sight (LOS) integration is also performed to determine the observable spectra through SDO/AIA. The numerical results show that the gradient of the coronal spectra is directly correlated with the chromosperic temperature configuration. As the chromospheric cavity size increases, the spectral gradient becomes shallower. When LOS integrations is performed, the resulting spectra demonstrate a broadband of excited frequencies that is correlated with the chromospheric cavity size. The broadband of excited frequencies becomes narrower as the chromospheric cavity size increases. These two results provide a potentially useful diagnostic for the chromospheric temperature profile by considering coronal velocity oscillations

Site effect variability of the Roio basin in the near-source area of the L'Aquila mainshock

In the frame of the microzonation studies of the April 6th, 2009 L'Aquila earthquake near-source area, we observed local seismic amplifications in the Roio area, a plain separated from L'Aquila city center by mount Luco. Six portable, digital instruments were deployed across the plain from the 15th of April until mid-May 2009. This array recorded up to 152 aftershocks. We analysed the ground motion from these events to determine relative site amplification within the plain and on the surrounding ridges. Horizontal over Vertical spectral ratio on noise data (HVSRN) and on aftershock recordings (HVEQ)as well as standard spectral ratio (SSR) show amplifications at 1.3 Hz and 4.0 Hz on the quaternary deposits. Seismic amplifications in the frequency range between 4 and 6 Hz were also observed on a carbonate ridge of Colle di Roio located on the northwest border of the plateau. A small amplification has also been noticed close to the top of mount Luco, another rocky site. This paper details the results we have obtained and addresses their interpretation.Submitted4.1. Metodologie sismologiche per l'ingegneria sismicaJCR Journalope

Seismic site response estimation in the near source region of the 2009 L'Aquila, Italy, Earthquake

To better estimate the seismic ground motion during the April 6th, 2009 earthquake in L'Aquila, we deployed temporary arrays in the near-source region. Several arrays have been successively set up in the Aterno valley's epicentral area and have recorded the aftershocks that followed the main shock, between April and September. The data has been processed in order to study the spectral ratios of the horizontal component of ground motion at the soil site and at a reference site, as well as the spectral ratio of the horizontal and the vertical movement at a single recording site. The results obtained confirm the presence of large amplification effects in both L'Aquila's historic centre and in the suburban areas. The resonance frequency has been found to be close to 0.6 Hz in downtown L'Aquila whereas the suburban areas show amplification at frequencies ranging from 2 Hz to 5 Hz.PublishedSkopje, Macedonia4.1. Metodologie sismologiche per l'ingegneria sismicaope

Nonlinear force-free magnetic field extrapolations: comparison of the Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithm

We compare the performance of two alternative algorithms which aim to construct a force-free magnetic field given suitable boundary conditions. For this comparison, we have implemented both algorithms on the same finite element grid which uses Whitney forms to describe the fields within the grid cells. The additional use of conjugate gradient and multigrid iterations result in quite effective codes. The Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithms both perform well for the reconstruction of a known analytic force-free field. For more arbitrary boundary conditions the Wheatland-Sturrock-Roumeliotis approach has some difficulties because it requires overdetermined boundary information which may include inconsistencies. The Grad-Rubin code on the other hand loses convergence for strong current densities. For the example we have investigated, however, the maximum possible current density seems to be not far from the limit beyond which a force free field cannot exist anymore for a given normal magnetic field intensity on the boundary.Comment: 21 pages, 13 figure