| Conventional linear stability analyses may fail for fluid systems with an indefinite free energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave-wave coupling, which leads to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially non-resonant systems to reach resonance by diffusion in phase space.; This is illustrated for a simple equilibrium involving cold counter-streaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, which occur generically for long enough wavelengths. Three-wave interactions which occur in isolated, but numerous, regions of parameter space can drive either decay instability or explosive instability. When the resonance for explosive growth is detuned, a stable region exists around the equilibrium point in phase space, while explosive growth occurs outside of a separatrix.; These interactions may be described exactly if only one resonance is considered, while multiple nonlinear terms make the Hamiltonian nonintegrable. Simple Hamiltonians of two and three degrees of freedom are studied numerically using symplectic integration algorithms, including an explicit algorithm derived using Lie algebraic methods. Two-wave and three-wave decay interactions lead to strongly chaotic motion, which destroys the separatrix bounding the stable region for near-resonant triplets. Phase space orbits experience slow diffusive growth to amplitudes sufficient for explosive instability, thus effectively reducing the critical amplitude. For Hamiltonians with more than two degrees of freedom, there is actually no critical amplitude for growth, because small perturbations may grow to arbitrary size via Arnold diffusion. It is observed numerically that this diffusion can be very slow for the smallest perturbations, although the actual diffusion rate is probably underestimated due to the simplicity of the model. |
