Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 331 / 3 / a

Under the convention $e^{i(kx-\omega t)}$, a mode's amplitude grows as $e^{{\omega }_{i}t}$. The linear stability analysis of real-wavenumber disturbances is therefore summarized byAt ${\omega }_{i,\max }=0$ there is marginal or neutral hydrodynamic stability: no normal mode grows exponentially, but some may fail to decay. If “linearly stable” includes neutral modes, its condition is ${\omega }_{i,\max }\le 0$. These modal statements do not rule out transient growth from non-normal modes or algebraic growth.

The fixed observer is the ray $V=0$. The physically selected zero-group velocity saddle defines the absolute wavenumber $k_0$ and absolute frequency ${\omega }_0$:For a smooth frequency branch with $D_{\omega }\ne 0$, the saddle condition is equivalently $D_k(k_0,{\omega }_0)=0$. The absolute growth rate isHere $k_0$ and ${\omega }_0$ can be complex. A physical saddle point is selected by the localized impulse response and deformation of its Fourier transform contour; an arbitrary algebraic stationary point need not determine absolute hydrodynamic instability.

For a temporally unstable base flow, the relevant saddle growth rate along a ray distinguishes two cases in the chosen observation frame:A convective hydrodynamic instability amplifies a travelling wave packet but lets a localized disturbance decay at any fixed point. An absolute hydrodynamic instability grows at a fixed point. The separating case $\sigma (0)=0$ is marginal in exponential rate and can have an algebraic prefactor. In a frame moving with $V$, replace $\sigma (0)$ by $\sigma (V)=\mathrm{Im}({\omega }_{\star }-V{k}_{\star })$: this distinction depends on the observation frame.