Metrics in Keplerian orbits quotient spaces

Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space \(\mathbb H\). Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into \(\mathbb R^n\), and a space of compact subsets of \(\mathbb H\) with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of \(\mathbb H\) and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in \(\mathbb H\) under coordinates change is discussed.     

Evidence of a shallow magma intrusion beneath the NE Rift system of Mt. Etna during 2013

Continuous GPS (CGPS) data, collected at Mt. Etna between April 2012 and October 2013, clearly define inflation/deflation processes typically observed before/after an eruption onset. During the inflationary process from May to October 2013, a particular deformation pattern localised in the upper North Eastern sector of the volcano suggests that a magma intrusion had occurred a few km away from the axis of the summit craters, beneath the NE Rift system. This is the first time that this pattern has been recorded by CGPS data at Mt. Etna. We believe that this inflation process might have taken place periodically at Mt. Etna and might be associated with the intrusion of batches of magma that are separate from the main feeding system. We provide a model to explain this unusual behaviour and the eruptive regime of this rift zone, which is characterised by long periods of quiescence followed by often dangerous eruptions in which vents can open at low elevation and thus threaten the villages in this sector of the volcano.     

Turbulence-induced steady streaming in an oscillating boundary layer: On the reliability of turbulence closure models

The accuracy of several closure models of the Reynolds-Averaged Navier–Stokes Equations in predicting the characteristics of an oscillating turbulent wall boundary layer is analyzed. The analysis involves four low Reynolds number k − ε models and a k − ω model and it is carried out by comparing the model results both with experimental data and with data obtained by a Direct Numerical Simulation (DNS) of the Navier–Stokes equations. The boundary layer is generated by a spatially constant time-oscillating pressure gradient given by the sum of two harmonic components characterized by angular frequencies Ω^∗ and 2Ω^∗ respectively, which generates a steady streaming because of the asymmetry of turbulence intensity during the cycle. Thus the results are relevant to the boundary layer at the bottom of nonlinear sea waves. The attention is therefore focused on the accuracy of the models in reproducing the period averaged profiles of the hydrodynamic characteristics of the steady streaming. The instantaneous quantities, such as time development of the wall shear stress, profiles of the streamwise velocity, Reynolds stresses and turbulent kinetic energy are also considered and analyzed. The results shows that a model can be judged better or worse than other models depending on the specific flow characteristic under investigation. However, an approach has been adopted which allowed to rank the models according to their accuracy in predicting the values of the hydrodynamic quantities involved in the present study.     

Experimental investigation on sea ripple evolution over sloping beaches

This paper reports on a wave flume experimental campaign carried out to investigate the appearance, the growth and the migration of small scale bedforms on a sloping sandy bed due to both regular and random waves. A Vectrino Profiler along with a structured light approach were used for velocity and morphodynamic measurements at two positions, one located above the horizontal bed, and the other one above the sloping beach. The velocity was computed by phase averaging the velocity measurements. Several velocity profiles were analyzed, identifying an offshore-directed steady current that extends from few centimeters above the bottom for all the analyzed water column. Ripple geometry was measured by a structured light approach and compared with that predicted by several models to shed light on the effects induced by the sloping beach on the shape and asymmetry. Along the sloping beach, the ripples appeared strongly asymmetric with the onshore half wavelengths smaller than the offshore ones. Finally, ripple geometry and migration triggered by regular waves were compared with those generated by random waves with comparable flow orbital amplitude showing a good agreement.     

A Joint Inversion of Ground Deformation and Focal Mechanisms Data for Magmatic Source Modelling

The paucity of geodetic data acquired on active volcanoes can make the understanding of modelling magmatic systems quite difficult. In this study, we propose a novel approach, which allows improving the parameter estimation of analytical models of magmatic sources (e.g., shape, depth, dimensions, volume change, etc.) by means of a joint inversion of surface ground deformation data and P-axes of focal plane solutions. The methodology is first verified against a synthetic dataset of surface deformation and strain within the medium, and then applied to real data from an unrest episode occurred before the May 13 2008 eruption at Mt. Etna (Italy). The main results clearly indicate the joint inversion improves the accuracy of the estimated source parameters by about 70 %. The statistical tests indicate that the source depth is the parameter with the highest increment of accuracy. In addition, a sensitivity analysis confirms that displacements data are more useful to constrain the pressure and the horizontal location of the source than its depth, while the P-axes better constrain the depth estimation.