Single-mode vortex as an elementary vortex geometric structure of an ideal fluid

A single-mode vortex is considered as an elementary solution to the equations of an ideal incompressible fluid in Arnold's geometric formulation 1. It is shown to be defined by a single harmonic mode and to represent a divergence-free velocity field with helical streamline geometry. In the Lie algebra of volume-preserving dif-feomorphisms, a single-mode vortex corresponds to an Abelian element whose commutator with itself is zero, ensuring the absence of nonlinear self-generation of modes and the stationarity of the solution 2. It is established that such vortices are geodesic trajectories on the group of diffeomorphisms and serve as fundamental building blocks of more complex vortex structures.