Hydrodynamic stability

Hydrodynamic stability studies whether small disturbances to a fluid base state decay, remain neutral, or grow. Linear normal modes reduce this question to eigenvalues or growth rates.

For a parallel inviscid base flow $U(z)$ and disturbance streamfunction $\varphi (z)e^{i\alpha (x-ct)}$, the Rayleigh equation is

An inviscid parallel shear flow can possess an exponentially growing normal mode only if $U^{\prime \prime }$ changes sign somewhere in the flow.

If an inviscid shear-flow mode has complex wave speed $c$ and $U_{-}\le U\le U_{+}$, then

Individually decaying nonorthogonal modes can interfere constructively and produce temporary energy growth. This mechanism is absent for an orthogonal eigenbasis of a normal evolution operator.

The energy-stability threshold is the parameter value below which a positive perturbation energy decreases monotonically for every disturbance. It can be lower than the linear instability threshold when non-normal transient growth is possible.

Symmetric instability occurs in rotating stratified flow when perturbations aligned with the basic current can extract energy from vertical shear. It is excluded when the Ertel potential vorticity has the stable sign; for constant gradients this gives a condition relating stratification, rotation, and horizontal buoyancy gradient.

The Eady model describes quasi-geostrophic baroclinic instability in a uniformly stratified layer with constant vertical shear, zero interior potential-vorticity gradient, and two rigid horizontal boundaries.

Eady instability arises when counterpropagating Rossby waves localized on the upper and lower boundaries phase-lock and amplify one another.

A boundary Rossby wave is supported by a buoyancy or potential-temperature anomaly on a rigid horizontal boundary. Its intrinsic propagation opposes the local basic flow in the Eady model.

Adding linear damping to one boundary of the Eady model makes the two boundary-wave phase speeds complex and can destabilize modes that were neutral without damping.

Dissipation-induced instability occurs when damping one component of a coupled system moves an eigenvalue into the growing half-plane. Selective damping can alter the phase relation that previously stabilized two interacting waves or oscillators.

For inviscid axisymmetric disturbances to a rotating flow, the Rayleigh discriminant measures the radial gradient of squared specific angular momentum.

An inviscid rotating flow is centrifugally stable to axisymmetric disturbances when $(r^2\Omega {)}^2$ is nondecreasing with radius. An outward decrease permits centrifugal instability.