On the algebra and groups of incompressible vortex dynamics

Abstract

An algebraic representation for 2D and 3D incompressible, inviscid fluid motion based on the continuous Nambu representation of Helmholtz vorticity equation is introduced. The Nambu brackets of conserved quantities generate a Lie algebra. Physically, we introduce matrix representations for the components of the linear momentum (2D and 3D), the circulation (2D) and the total flux of vorticity (3D). These quantities form the basis of the vortex-Heisenberg Lie algebra. Applying the matrix commutator to the basis matrices leads to the same physical relations as the Nambu bracket for this quantities expressed classically as functionals. Using the matrix representation of the Lie algebra we derive the matrix and vector representations for the nilpotent vortex-Heisenberg groups that we denote by VH(2) and VH(3). It turns out that VH(2) is a covering group of the classical Heisenberg group for mass point dynamics. VH(3) can be seen as central extension of the abelian group of translations. We further introduce the Helmholtz vortex group V(3), where additionally the angular momentum is included. Regarding application-oriented aspects, the novel matrix representation might be useful for numerical investigations of the group, whereas the vector representation of the group might provide a better process-related understanding of vortex flows.