Motion of an ideal fluid is represented as geodesics on the group of all volume-preserving diffeomorphisms. An explicit form of the geodesic equation is presented for the fluid flow on a three-torus Riemannian connection, commutator and curvature tensor are given explicitly and applied to a couple of simple flows with the Beltrami property. It is found that the curvature is non-positive for the section of two ABC flows with different values of the constants (A, B and C). The study is an extension of the Arnold's results (1989) in the two-dimensional case to three-dimensional fluid motions.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)