Abstract:We show obstructions to the existence of a coclosed G -structure on a Lie algebra g of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from g to a six-dimensional Lie algebra h, with the kernel contained in the center of g, then any coclosed G -structure on g induces a closed and stable three form on h that defines an almost complex structure on h. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed G -structures. We also prove that each one of these Lie algebras has a coclosed G -structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed G -structures. The existence of contact metric structures is also studied.

We find explicit solutions of the Laplacian coflow of G 2 −structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existence.

We study the Laplacian coflow and the modified Laplacian coflow of G 2 -structures on the 7-dimensional Heinseberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval (−∞, T ), with 0 < T < +∞. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while in other cases the solution is defined for every positive time.

Submesoscale eddies play an important role in the energy transfer from the mesoscale down to the dissipative range, as well as in tracer transport. They carry inorganic matter, nutrients and biomass; in addition, they may act as pollutant conveyors. However, synoptic observations of these features need high resolution sampling, in both time and space, making their identification challenging. Therefore, HF coastal radar were and are successfully used to accurately identify, track and describe them. In this paper we tested two already existing algorithms for the automated detection of submesoscale eddies. We applied these algorithms to HF radar velocity fields measured by a network of three radar systems operating in the Gulf of Naples. Both methods showed shortcomings, due to the high non-geostrophy of the observed currents. For this reason we developed a third, novel algorithm that proved to be able to detect highly asymmetrical eddies, often not properly identified by the previous ones. We used the results of the application of this algorithm to estimate the eddy boundary profiles and the eddy spatial distribution.

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