We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

Larmor size transient structures with depletions as large as 99% of ambient solar wind density levels occur commonly upstream of Earth's coUisionless bow shock. These "density holes" have a mean duration of-17.9 ± 10.4s but holes as short as 4s have been observed. The average fractional density depletion (5n/n) inside the holes is-0.68 ± 0.14. The density of the upstream edge moving in the sunward direction can be enhanced by five or more times the solar wind density. Particle distributions show the steepened edge can behave like a shock, and measured local field geometries and Mach number support this view. Similarly shaped magnetic holes accompany the density holes indicating strong coupling between fields and particles. The density holes are only observed with upstream particles, suggesting that back-streaming particles interacting with the solar wind are important.

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