Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 331 / 4 / b / vi

Both eigenvalues of the linear generator, $-1/R$ and $-1/(4R)$, are negative. Nevertheless the non-normal matrix transfers the slowly decaying second component into the first before viscous decay wins. Modal decay coexists with energy amplification of order $R^2$ over a time of order $R$.

This illustrates transient growth from non-normal modes: large Reynolds number permits much greater disturbance amplification even though the linear stability analysis finds no unstable eigenvalue. The energy-neutral rotational nonlinearity can redirect the amplified state and affect its subsequent evolution. The linear gain alone establishes neither sustained nonlinear instability nor a nonlinear transition threshold.