Clarke subgradients of stratifiable functions

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure

Improvement of solar cycle prediction: Plateau of solar axial dipole moment

Aims. We report the small temporal variation of the axial dipole moment near the solar minimum and its application to the solar cycle prediction by the surface flux transport (SFT) model. Methods. We measure the axial dipole moment using the photospheric synoptic magnetogram observed by the Wilcox Solar Observatory (WSO), the ESA/NASA Solar and Heliospheric Observatory Michelson Doppler Imager (MDI), and the NASA Solar Dynamics Observatory Helioseismic and Magnetic Imager (HMI). We also use the surface flux transport model for the interpretation and prediction of the observed axial dipole moment. Results. We find that the observed axial dipole moment becomes approximately constant during the period of several years before each cycle minimum, which we call the axial dipole moment plateau. The cross-equatorial magnetic flux transport is found to be small during the period, although the significant number of sunspots are still emerging. The results indicates that the newly emerged magnetic flux does not contributes to the build up of the axial dipole moment near the end of each cycle. This is confirmed by showing that the time variation of the observed axial dipole moment agrees well with that predicted by the SFT model without introducing new emergence of magnetic flux. These results allows us to predict the axial dipole moment in Cycle 24/25 minimum using the SFT model without introducing new flux emergence. The predicted axial dipole moment of Cycle 24/25 minimum is 60--80 percent of Cycle 23/24 minimum, which suggests the amplitude of Cycle 25 even weaker than the current Cycle 24. Conclusions. The plateau of the solar axial dipole moment is an important feature for the longer prediction of the solar cycle based on the SFT model.Comment: 5 pages, 3 figures, accepted for publication in A&A Lette

Self-Consistent MHD Modeling of a Coronal Mass Ejection, Coronal Dimming, and a Giant Cusp-Shaped Arcade Formation

We performed magnetohydrodynamic simulation of coronal mass ejections (CMEs) and associated giant arcade formations, and the results suggested new interpretations of observations of CMEs. We performed two cases of the simulation: with and without heat conduction. Comparing between the results of the two cases, we found that reconnection rate in the conductive case is a little higher than that in the adiabatic case and the temperature of the loop top is consistent with the theoretical value predicted by the Yokoyama-Shibata scaling law. The dynamical properties such as velocity and magnetic fields are similar in the two cases, whereas thermal properties such as temperature and density are very different.In both cases, slow shocks associated with magnetic reconnectionpropagate from the reconnection region along the magnetic field lines around the flux rope, and the shock fronts form spiral patterns. Just outside the slow shocks, the plasma density decreased a great deal. The soft X-ray images synthesized from the numerical results are compared with the soft X-ray images of a giant arcade observed with the Soft X-ray Telescope aboard {\it Yohkoh}, it is confirmed that the effect of heat conduction is significant for the detailed comparison between simulation and observation. The comparison between synthesized and observed soft X-ray images provides new interpretations of various features associated with CMEs and giant arcades.Comment: 39 pages, 18 figures. Accepted for publication in the Astrophysical Journal. The PDF file with high resplution figures can be downloaded from http://www.kwasan.kyoto-u.ac.jp/~shiota/study/ApJ62426.preprint.pdf

Three Dimensional Structure and Energy Balance of a Coronal Mass Ejection

The Ultraviolet Coronagraph Spectrometer (UVCS) observed Doppler shifted material of a partial Halo Coronal Mass Ejection (CME) on December 13 2001. The observed ratio of [O V]/O V] is a reliable density diagnostic important for assessing the state of the plasma. Earlier UVCS observations of CMEs found evidence that the ejected plasma is heated long after the eruption. We have investigated the heating rates, which represent a significant fraction of the CME energy budget. The parameterized heating and radiative and adiabatic cooling have been used to evaluate the temperature evolution of the CME material with a time dependent ionization state model. The functional form of a flux rope model for interplanetary magnetic clouds was also used to parameterize the heating. We find that continuous heating is required to match the UVCS observations. To match the O VI-bright knots, a higher heating rate is required such that the heating energy is greater than the kinetic energy. The temperatures for the knots bright in Ly$\alpha$ and C III emission indicate that smaller heating rates are required for those regions. In the context of the flux rope model, about 75% of the magnetic energy must go into heat in order to match the O VI observations. We derive tighter constraints on the heating than earlier analyses, and we show that thermal conduction with the Spitzer conductivity is not sufficient to account for the heating at large heights.Comment: 40 pages, 16 figures, accepted for publication in ApJ For associated mpeg file, please see https://www.cora.nwra.com/~jylee/mpg/f5.mp

Polar Field Reversal Observations with Hinode

We have been monitoring yearly variation in the Sun's polar magnetic fields with the Solar Optical Telescope aboard {\it Hinode} to record their evolution and expected reversal near the solar maximum. All magnetic patches in the magnetic flux maps are automatically identified to obtain the number density and magnetic flux density as a function of th total magnetic flux per patch. The detected magnetic flux per patch ranges over four orders of magnitude ($10^{15}$ -- $10^{20}$ Mx). The higher end of the magnetic flux in the polar regions is about one order of magnitude larger than that of the quiet Sun, and nearly that of pores. Almost all large patches ($\geq 10^{18}$ Mx) have the same polarity, while smaller patches have a fair balance of both polarities. The polarity of the polar region as a whole is consequently determined only by the large magnetic concentrations. A clear decrease in the net flux of the polar region is detected in the slow rising phase of the current solar cycle. The decrease is more rapid in the north polar region than in the south. The decrease in the net flux is caused by a decrease in the number and size of the large flux concentrations as well as the appearance of patches with opposite polarity at lower latitudes. In contrast, we do not see temporal change in the magnetic flux associated with the smaller patches ($< 10^{18}$ Mx) and that of the horizontal magnetic fields during the years 2008--2012.Comment: 21 pages, 7 figures. Accepted for publication in Ap