Alliance of transient growth and sub-criticality in the Lamb-Oseen vortex: an amplitude equation approach*

We analytically derive an amplitude equation for the weakly nonlinear evolution of the linearly most amplified response of a non-normal dynamical system. The development generalizes

a recently proposed operator exponential perturbation approach, in that the base flow here arbitrarily depends on time. Applied to the two-dimensional Lamb-Oseen vortex, the amplitude equation successfully predicts the nonlinearities to weaken or reinforce the transient gain. In particular, the minimum amplitude of the linear optimal initial condition required for a sub-critical bifurcation to happen in this flow is found to decay with the Reynolds number, which is confirmed by direct numerical simulations. The simplicity of the amplitude equation and the link made with the sensitivity formula suggests a physical interpretation of nonlinear effects, in light of existing work on Landau damping and on shear instabilities. The amplitude equation also quantifies the respective contributions of the second harmonic and the spatial mean flow distortion in the nonlinear modification of the gain.