Clebsch maps encode velocity fields through functions. These functions contain valuable information about the velocity field. For example, closed integral curves of the associated vorticity field are level lines of the vorticity Clebsch map. This makes Clebsch maps useful for visualization and fluid dynamics analysis. Additionally they can be used in the context of simulations to enhance flows through the introduction of subgrid vorticity. In this paper we study spherical Clebsch maps, which are particularly attractive. Elucidating their geometric structure, we show that such maps can be found as minimizers of a non-linear Dirichlet energy. To illustrate our approach we use a number of benchmark problems and apply it to numerically given flow fields. Code and a video can be found in the ACM Digital Library (and in this project page).
This work was supported in part by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics." Additional support was provided by SideFX software. The Bunny model is courtesy Stanford Computer Graphics laboratory. The hummingbird flow data and photogrammetrically acquired bird geometry courtesy Haibo Dong, Flow Simulation Research Group, University of Virginia. The Delta Wing data set courtesy NASA Advanced Supercomputing Division.