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We propose a novel method to extract hierarchies of vortex filaments from given three-dimensional flow velocity fields. We call these collections of filaments Hierarchical Vorticity Skeletons (HVS). They extract multi-scale information from the input velocity field, which is not possible with any previous filament extraction approach. Once computed, these HVSs provide a powerful mechanism for data compression and a very natural way for modifying flows. The data compression rates for all presented examples are above 99%. Employing our skeletons for flow modification has several advantages over traditional approaches. Most importantly, they reduce the complexity of three-dimensional fields to one-dimensional lines and, make complex fluid data more accessible for changing defining features of a flow. The strongly reduced HVS dataset still carries the main characteristics of the flow. Through the hierarchy we can capture the main features of different scales in the flow and by that provide a level of detail control. In contrast to previous work, we present a fully automated pipeline to robustly decompose dense velocities into filaments.