Concept

Critical layer — where it appears

The level in a shear flow where the flow's speed equals a wave's phase speed, so that the wave and the fluid there travel together. Energy is exchanged between wave and mean flow at that level, and it is where a disturbance can grow or be absorbed.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached.

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

turbulence · Instability

A thin interface stays unstable far past a quarter. The fastest growth rate of any disturbance against the bulk Richardson number J, on a logarithmic scale, for a density interface as thick as the shear and two, two and a half and three times thinner. The matched layer's billow dies at a quarter, as Miles' theorem requires. Two times thinner, the billow dies sooner, its last growth at J = 0.12, and nothing replaces it. Three times thinner, the billow's last is at 0.08 and a travelling wave takes over: 0.0335 at a quarter, 0.0109 at one and 0.00629 at 1.3, falling steadily with no threshold in the range solved. At 2.5 the waves are weaker and reach 0.001 at 1.3.

A thin interface keeps its waves past a quarter

Miles' quarter rules a shear layer whose density changes over the same depth as its velocity. Make the density interface three times thinner and the stationary billow dies early, but a pair of travelling waves takes its place and is still growing at five times the quarter. No theorem is broken: at the edges of the shear, where the waves draw their energy, the local Richardson number has fallen to nothing.

turbulence · Stratification

Named alongside it

The objects these essays reach for when they reach for this one.

Eigenvalue Linear stability Phase speed Buoyancy Dissipation Inflection point Kelvin helmholtz Model limit Rayleigh equation Reynolds number Reynolds stress Richardson number

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