 Hydrodynamic stability

Hydrodynamic stability studies how a fluid flow responds to a small disturbance of a specified base state. The base state, boundary conditions and measure of disturbance size are part of the problem. Linearizing the Navier-Stokes equation gives a perturbation evolution equation; a normal mode proportional to $e^{st}$ grows exponentially when $\mathrm{Re}s>0$. For a viscous parallel shear flow, the Orr-Sommerfeld equation and Squire equation describe this modal problem.

A complete account also asks about transient growth. Stable eigenvalues do not guarantee that perturbation kinetic energy decreases at every instant; even a non-normal matrix with decaying modes can exhibit transient growth. The lift-up effect supplies a concrete mechanism: cross-stream velocity displaces fluid between regions of different mean streamwise momentum, producing streamwise velocity proportional to time in the ideal inviscid calculation. A finite disturbance can consequently become much larger before eventual decay.

Different physical mechanisms require different stability tests. The Rayleigh-Taylor instability releases gravitational potential energy from an unfavorable density arrangement, while baroclinic instability draws on the available potential energy of a stratified sheared state. A linear instability identifies initial amplification; predicting the eventual flow requires nonlinear dynamics and its additional constraints.