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

Error Analysis regarding the calculation of NLFF Field

Field extrapolation is an alternative method to study chromospheric and coronal magnetic fields. In this paper, two semi-analytical solutions of force- free fields (Low and Lou, 1990) have been used to study the errors of nonlin- ear force-free (NLFF) fields based on force-free factor alpha. Three NLFF fields are extrapolated by approximate vertical integration (AVI) Song et al. (2006), boundary integral equation (BIE) Yan and Sakurai (2000) and optimization (Opt.) Wiegelmann (2004) methods. Compared with the first semi-analytical field, it is found that the mean values of absolute relative standard deviations (RSD) of alpha along field lines are about 0.96-1.05, 0.94-1.07 and 0.46-0.72 for AVI, BIE and Opt. fields, respectively. While for the second semi-analytical field, they are about 0.80-1.02, 0.63-1.34 and 0.33-0.55 for AVI, BIE and Opt. fields, respectively. As for the analytical field, the calculation error of hjRSDji is about 0.1 {\guillemotright} 0.2. It is also found that RSD does not apparently depend on the length of field line. These provide the basic estimation on the deviation of extrapolated field obtained by proposed methods from the real force-free field.Comment: 22 pages, 13 figures, "Accepted for publication in Astrophysics & Space Science

Small group interventions for children aged 5-9 years old with mathematical learning difficulties

The research related to educational interventions for children with mathematical learning difficulties has been increasing steadily. In this chapter I focus on small group interventions for children aged 5–9 years old with learning difficulties in mathematics. First, I describe the important issues: (1) who are the children having problems in mathematics, (2) what do we mean with (special) education intervention, (3) what does Responsiveness to Intervention mean, and (4) what intervention features have been found effective for children aged 5–9 years with learning difficulties in mathematics. Then, I describe the research and developmental work that has been done in Finland on designing web services which provide evidence-based information and materials for educators. The two web services are LukiMat and ThinkMath. Together, these two web services include the knowledge base, assessment batteries and intervention tools to be used in relation to mathematical learning difficulties in the age group 5–9 years.Peer reviewe