Extending the range of error estimates for radial approximation in Euclidean space and on spheres

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.Comment: 10 page

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L∞(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates

Turbulence in the Solar Atmosphere: Manifestations and Diagnostics via Solar Image Processing

Intermittent magnetohydrodynamical turbulence is most likely at work in the magnetized solar atmosphere. As a result, an array of scaling and multi-scaling image-processing techniques can be used to measure the expected self-organization of solar magnetic fields. While these techniques advance our understanding of the physical system at work, it is unclear whether they can be used to predict solar eruptions, thus obtaining a practical significance for space weather. We address part of this problem by focusing on solar active regions and by investigating the usefulness of scaling and multi-scaling image-processing techniques in solar flare prediction. Since solar flares exhibit spatial and temporal intermittency, we suggest that they are the products of instabilities subject to a critical threshold in a turbulent magnetic configuration. The identification of this threshold in scaling and multi-scaling spectra would then contribute meaningfully to the prediction of solar flares. We find that the fractal dimension of solar magnetic fields and their multi-fractal spectrum of generalized correlation dimensions do not have significant predictive ability. The respective multi-fractal structure functions and their inertial-range scaling exponents, however, probably provide some statistical distinguishing features between flaring and non-flaring active regions. More importantly, the temporal evolution of the above scaling exponents in flaring active regions probably shows a distinct behavior starting a few hours prior to a flare and therefore this temporal behavior may be practically useful in flare prediction. The results of this study need to be validated by more comprehensive works over a large number of solar active regions.Comment: 26 pages, 7 figure

Magnetic Energy and Helicity Budgets in the Active-Region Solar Corona. I. Linear Force-Free Approximation

We self-consistently derive the magnetic energy and relative magnetic helicity budgets of a three-dimensional linear force-free magnetic structure rooted in a lower boundary plane. For the potential magnetic energy we derive a general expression that gives results practically equivalent to those of the magnetic Virial theorem. All magnetic energy and helicity budgets are formulated in terms of surface integrals applied to the lower boundary, thus avoiding computationally intensive three-dimensional magnetic field extrapolations. We analytically and numerically connect our derivations with classical expressions for the magnetic energy and helicity, thus presenting a so-far lacking unified treatment of the energy/helicity budgets in the constant-alpha approximation. Applying our derivations to photospheric vector magnetograms of an eruptive and a noneruptive solar active regions, we find that the most profound quantitative difference between these regions lies in the estimated free magnetic energy and relative magnetic helicity budgets. If this result is verified with a large number of active regions, it will advance our understanding of solar eruptive phenomena. We also find that the constant-alpha approximation gives rise to large uncertainties in the calculation of the free magnetic energy and the relative magnetic helicity. Therefore, care must be exercised when this approximation is applied to photospheric magnetic field observations. Despite its shortcomings, the constant-alpha approximation is adopted here because this study will form the basis of a comprehensive nonlinear force-free description of the energetics and helicity in the active-region solar corona, which is our ultimate objective.Comment: 44 pages, 8 figures, 2 tables. The Astrophysical Journal, in pres

Solar Flares as Cascades of Reconnecting Magnetic Loops

A model for the solar coronal magnetic field is proposed where multiple directed loops evolve in space and time. Loops injected at small scales are anchored by footpoints of opposite polarity moving randomly on a surface. Nearby footpoints of the same polarity aggregate, and loops can reconnect when they collide. This may trigger a cascade of further reconnection, representing a solar flare. Numerical simulations show that a power law distribution of flare energies emerges, associated with a scale free network of loops, indicating self-organized criticality.Comment: 4 pages, 4 figures, To be published in Phys. Rev. Let

Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control

Restoring tibiofemoral alignment during ACL reconstruction results in better knee biomechanics

"Published online: 24 October 2017"PURPOSE: Anterior cruciate ligament (ACL) reconstruction (ACLR) aims to restore normal knee joint function, stability and biomechanics and in the long term avoid joint degeneration. The purpose of this study is to present the anatomic single bundle (SB) ACLR that emphasizes intraoperative correction of tibiofemoral subluxation that occurs after ACL injury. It was hypothesized that this technique leads to optimal outcomes and better restoration of pathological tibiofemoral joint movement that results from ACL deficiency (ACLD). METHODS: Thirteen men with unilateral ACLD were prospectively evaluated before and at a mean follow-up of 14.9 (SD = 1.8) months after anatomic SB ACLR with bone patellar tendon bone autograft. The anatomic ACLR replicated the native ACL attachment site anatomy and graft orientation. Emphasis was placed on intraoperative correction of tibiofemoral subluxation by reducing anterior tibial translation (ATT) and internal tibial rotation. Function was measured with IKDC, Lysholm and the Tegner activity scale, ATT was measured with the KT-1000 arthrometer and tibial rotation (TR) kinematics were measured with 3Dmotion analysis during a high-demand pivoting task. RESULTS: The results showed significantly higher TR of the ACL-deficient knee when compared to the intact knee prior to surgery (12.2° ± 3.7° and 10.7° ± 2.6° respectively, P = 0.014). Postoperatively, the ACLR knee showed significantly lower TR as compared to the ACL-deficient knee (9.6°±3.1°, P = 0.001) but no difference as compared to the control knee (n.s.). All functional scores were significantly improved and ATT was restored within normal values (P < 0.001). CONCLUSIONS: Intraoperative correction of tibiofemoral subluxation that results after ACL injury is an important step during anatomic SB ACLR. The intraoperative correction of tibiofemoral subluxation along with the replication of native ACL anatomy results in restoration of rotational kinematics of ACLD patients to normal levels that are comparable to the control knee. These results indicate that the reestablishment of tibiofemoral alignment during ACLR may be an important step that facilitates normal knee kinematics postoperatively. LEVEL OF EVIDENCE: Level II, prospective cohort study.The authors gratefully acknowledge the funding support from the Hellenic Association of Orthopaedic Surgery and Traumatology (HAOST-EEXOT)info:eu-repo/semantics/publishedVersio

Unusual Stokes V profiles during flaring activity of a delta sunspot

We analyze a set of full Stokes profile observations of the flaring active region NOAA 10808 recorded with the Vector-Spectromagnetograph (VSM) of the SOLIS facility. We aim to quantify transient and permanent changes in the magnetic field and velocity field. The results are put in context with MDI magnetograms and reconstructed RHESSI X-ray images. We find signs of restructuring of the photospheric magnetic field during the flare close to the polarity inversion line (PIL) at the flaring site. At two locations in the umbra we encounter strong fields (approx. 3 kG), as inferred from the Stokes I profiles which, however, exhibit a low polarization signal. During the flare we observe in addition asymmetric Stokes V profiles at one of these sites. The asymmetric Stokes V profiles appear co-spatial and co-temporal with a strong apparent polarity reversal observed in MDI-magnetograms and a chromospheric hard X-ray source. The two-component atmosphere fits of the asymmetric Stokes profiles result in line-of-sight velocity differences in the range of approx. 12km/s to 14 km/s between the two components in the photosphere. Another possibility is that local atmospheric heating is causing the observed asymmetric Stokes V profile shape. In either case our analysis shows that a very localized patch of approx. 5 arcsec in the photospheric umbra, co-spatial with a flare footpoint, exhibits a sub-resolution fine structure.Comment: 13 pages, 10 figures, 1 tabl

Magnetic Flux Cancellation in Ellerman Bombs

Ellerman Bombs (EBs) are often found to be co-spatial with bipolar photospheric magnetic fields. We use Hα imaging spectroscopy along with Fe I 6302.5 Å spectropolarimetry from the Swedish 1 m Solar Telescope (SST), combined with data from the Solar Dynamic Observatory, to study EBs and the evolution of the local magnetic fields at EB locations. EBs are found via an EB detection and tracking algorithm. Using NICOLE inversions of the spectropolarimetric data, we find that, on average, (3.43 ± 0.49) × 1024 erg of stored magnetic energy disappears from the bipolar region during EB burning. The inversions also show flux cancellation rates of 1014–1015 Mx s−1 and temperature enhancements of 200 K at the detection footpoints. We investigate the near-simultaneous flaring of EBs due to co-temporal flux emergence from a sunspot, which shows a decrease in transverse velocity when interacting with an existing, stationary area of opposite polarity magnetic flux, resulting in the formation of the EBs. We also show that these EBs can be fueled further by additional, faster moving, negative magnetic flux regions

A Helicity-Based Method to Infer the CME Magnetic Field Magnitude in Sun and Geospace: Generalization and Extension to Sun-Like and M-Dwarf Stars and Implications for Exoplanet Habitability

Patsourakos et al. (Astrophys. J. 817, 14, 2016) and Patsourakos and Georgoulis (Astron. Astrophys. 595, A121, 2016) introduced a method to infer the axial magnetic field in flux-rope coronal mass ejections (CMEs) in the solar corona and farther away in the interplanetary medium. The method, based on the conservation principle of magnetic helicity, uses the relative magnetic helicity of the solar source region as input estimates, along with the radius and length of the corresponding CME flux rope. The method was initially applied to cylindrical force-free flux ropes, with encouraging results. We hereby extend our framework along two distinct lines. First, we generalize our formalism to several possible flux-rope configurations (linear and nonlinear force-free, non-force-free, spheromak, and torus) to investigate the dependence of the resulting CME axial magnetic field on input parameters and the employed flux-rope configuration. Second, we generalize our framework to both Sun-like and active M-dwarf stars hosting superflares. In a qualitative sense, we find that Earth may not experience severe atmosphere-eroding magnetospheric compression even for eruptive solar superflares with energies ~ 10^4 times higher than those of the largest Geostationary Operational Environmental Satellite (GOES) X-class flares currently observed. In addition, the two recently discovered exoplanets with the highest Earth-similarity index, Kepler 438b and Proxima b, seem to lie in the prohibitive zone of atmospheric erosion due to interplanetary CMEs (ICMEs), except when they possess planetary magnetic fields that are much higher than that of Earth.Comment: http://adsabs.harvard.edu/abs/2017SoPh..292...89