2005/12/31 by John D. Gibbon, J D Gibbon, Darryl D. Holm +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #nlin.CD
paper · pdf · doi:10.1088/0951-7715/19/8/011
14 pages, 3 figures, revised abstract, To appear in Nonlinearity
arxiv created 2006/06/22 · openalex publication_date 2006/07/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet–Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.