Geometric stability of stationary Euler flows

Abstract

ABSTRACT Geometric stability theory is developed as an analogue of structural stability in physical space. Two steady flows are said to be geometrically equivalent if they have the same streamline pattern with velocities satisfying . A stationary solution to the Euler equations is non-unique if there exists a geometrically equivalent solution with horizontally varying F, in which case its geometric structure is stable and permits non-proportional velocity change. An Euler flow without such an equivalent solution is unique and geometrically unstable. Analysis of pseudo-plane flows shows that only constant-speed flows, specifically straightline jet and vertical-aligned circular vortex, are geometrically stable. The only stable flow with closed streamlines is vertical-aligned circular vortex, which provides a stability explanation for the phenomenon of vortex alignment and axisymmetrisation. A series of polynomial and nonpolynomial Euler solutions is used to validate the generic instability of pseudo-plane ideal flows. The quadratic solutions indicate that the pressure field has dynamic multiplicity and cannot be used as a proxy for geometric analysis.