A mostly linear model of transition to turbulence

A simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part of the model is stable but highly non-normal, so that certain inputs experience a great deal of growth before they eventually decay. The nonlinear terms of the model contribute no energy growth, but recycle some of the linear outputs into inputs, closing a feedback loop and allowing initially small solutions to ‘‘bootstrap’’ to a much larger amplitude. Although different choices of parameters in the nonlinearity lead to a variety of long-term behaviors, the bootstrapping process is essentially independent of the details of the nonlinearity and varies predictably with the Reynolds number. The bootstrapping scenario demonstrated by this model is the basis of some recent explanations for the failure of classical hydrodynamic stability analysis to predict the onset of turbulence in certain flow configurations.