Evolution of the flow field in decaying active regions, Transition from a moat flow to a supergranular flow

We investigate the evolution of the horizontal flow field around sunspots during their decay by analysing its extension and horizontal velocity around eight spots using SDO/HMI Doppler maps. By assuming a radially symmetrical flow field, the applied analysis method determines the radial dependence of the azimuthally averaged flow field. For comparison, we studied the flow in supergranules using the same technique. All investigated, fully fledged sunspots are surrounded by a flow field whose horizontal velocity profile decreases continuously from 881 m s$^{-1}$ at 1.1 Mm off the spot boundary, down to 199 m s$^{-1}$ at a mean distance of 11.9 Mm to that boundary. Once the penumbra is fully dissolved, however, the velocity profile of the flow changes: The horizontal velocity increases with increasing distance to the spot boundary until a maximum value of about 398 m s$^{-1}$ is reached. Then, the horizontal velocity decreases for farther distances to the spot boundary. In supergranules, the horizontal velocity increases with increasing distance to their centre up to a mean maximum velocity of 355 m s$^{-1}$. For larger distances, the horizontal velocity decreases. We thus find that the velocity profile of naked sunspots resembles that of supergranular flows. The evolution of the flow field around individual sunspots is influenced by the way the sunspot decays and by the interplay with the surrounding flow areas. Observations of the flow around eight decaying sunspots suggest that as long as penumbrae are present, sunspots with their moat cell are embedded in network cells. The disappearance of the penumbra (and consequently the moat flow) and the competing surrounding supergranular cells, both have a significant role in the evolution of the flow field: The moat cell transforms into a supergranule, which hosts the remaining naked spot.Comment: accepted for publication in A&A, 11 pages, 6 figures, 3 tables; appendix with 9 figures and 8 online movie

Geodesics for a class of distances in the space of probability measures

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savar e. We first prove the existence of a potential function and then give necessary and suffi cient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves

Infinite-horizon problems under periodicity constraint

We study so{\`u}e infinite-horizon optimization problems on spaces of periodic functions for non periodic Lagrangians. The main strategy relies on the reduction to finite horizon thanks in the introduction of an avering operator.We then provide existence results and necessary optimality conditions in which the corresponding averaged Lagrangian appears

Weighted interpolation inequalities: a perturbation approach

We study optimal functions in a family of Caffarelli-Kohn-Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime

Optimal transportation for the determinant

Among $\R^3$-valued triples of random vectors $(X,Y,Z)$ having fixed marginal probability laws, what is the best way to jointly draw $(X,Y,Z)$ in such a way that the simplex generated by $(X,Y,Z)$ has maximal average volume? Motivated by this simple question, we study optimal transportation problems with several marginals when the objective function is the determinant or its absolute value

Magnetic properties of a long-lived sunspot - Vertical magnetic field at the umbral boundary

Context. In a recent statistical study of sunspots in 79 active regions, the vertical magnetic field component $B_\text{ver}$ averaged along the umbral boundary is found to be independent of sunspot size. The authors of that study conclude that the absolute value of $B_\text{ver}$ at the umbral boundary is the same for all spots. Aims. We investigate the temporal evolution of $B_\text{ver}$ averaged along the umbral boundary of one long-lived sunspot during its stable phase. Methods. We analysed data from the HMI instrument on-board SDO. Contours of continuum intensity at $I_\text{c}=0.5I_\text{qs}$, whereby $I_\text{qs}$ refers to the average over the quiet sun areas, are used to extract the magnetic field along the umbral boundary. Projection effects due to different formation heights of the Fe I 617.3 nm line and continuum are taken into account. To avoid limb artefacts, the spot is only analysed for heliocentric angles smaller than $60^{\circ}$. Results. During the first disc passage, NOAA AR 11591, $B_\text{ver}$ remains constant at 1693 G with a root-mean-square deviation of 15 G, whereas the magnetic field strength varies substantially (mean 2171 G, rms of 48 G) and shows a long term variation. Compensating for formation height has little influence on the mean value along each contour, but reduces the variations along the contour when away from disc centre, yielding a better match between the contours of $B_\text{ver}=1693$ G and $I_\text{c}=0.5I_\text{qs}$. Conclusions. During the disc passage of a stable sunspot, its umbral boundary can equivalently be defined by using the continuum intensity $I_\text{c}$ or the vertical magnetic field component $B_\text{ver}$. Contours of fixed magnetic field strength fail to outline the umbral boundary.Comment: accepted for publication in A&A; v2 minor edit, correcting statement regarding one citatio