Stability theory and hamiltonian dynamics in the Euler ideal fluid equations

Abstract

The study of shear flow steady states has led to a wealth of research in the field of fluid dynamics. By studying shear flows, we can understand how a fluid behaves and how coherent structures arise. We primarily study the stability of shear flows in the Euler equations. The Euler equations describe the dynamics of an ideal fluid which is incompressible, inviscid, and experiences no external forces. We study the family of shear flows of the Euler equations with vorticity of the form Ω(x,y)=cos(κxpxx+κypyy) on a two-dimensional periodic domain of size [0,2π/κx)×[0,2π/κy), and formulate this as a Poisson system. We prove that if py=0 and κx|px|<κy, the flow is linearly stable. If (κx2px2+κx2px2)1/2>(3+2×31/2 )/2, we prove the flow is nonlinearly unstable. We discuss the full spectrum of the linearisation of shear flows and a related Jacobi problem. We prove analogous stability results in a known Poisson structure preserving truncation and discuss the qualitative differences. We extend a previously known Poisson integrator for this truncation of the Euler equations to a general two-dimensional periodic domain. The Euler equations on a three-dimensional periodic domain are less well-understood. In this domain we formulate the dynamics in terms of the vorticity Fourier modes. This is then used to study shear flows and prove similar stability results as for the two-dimensional case. The linearised equations split into subsystems which have equivalent dynamics to those of the two-dimensional subsystems. We prove the existence of a family of linearly stable shear flows, and another of linearly unstable shear flows. For a dense set of parameter values, the linearised system has a nilpotent part. This is linked to nonnormality and indicates a transition to turbulence. We formulate the Euler equations on a three-dimensional periodic domain as a Poisson system. We finally present some numerical results demonstrating and exploring the results of this thesis.