Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Buoyancy frequency — where it appears

The rate at which a parcel displaced vertically in a stratified fluid oscillates about its equilibrium level, N with N² = −(g/ρ)dρ/dz. It is a property of the fluid rather than of any motion, and it sets the highest frequency an internal wave can have.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A cross at 30.0° for every wavelength it makes. A body oscillating at ω = 0.5N in a stratified fluid, and the four beams along which its energy leaves. The angle is arccos(ω/N) from the vertical — 30.0 degrees from the horizontal here — and it is the same for every wavelength the body excites, because the dispersion relation has no length in it. The short strokes are the crests, which lie along the beams rather than across them: the phase advances perpendicular to the energy, and the two are exactly at right angles. Raise the frequency and the cross closes towards the vertical; reach ω = N and it shuts entirely, because nothing above the buoyancy frequency propagates.

The number that stops the mixing

A fluid whose density falls with height resists being stirred, and the resistance has a threshold at exactly one quarter, from an energy balance with no fluid mechanics in it. The waves such a fluid carries are stranger still — their frequency decides the direction they travel in and says nothing about their wavelength.

turbulence · Stratification Five numbers, one flow, one name. A hyperbolic-tangent shear layer with a matching density profile, measured five ways, each of which appears in the literature as "the Richardson number". The minimum gradient value is what Miles' and Howard's theorem is about and is the only one the quarter belongs to. The bulk numbers depend on which thickness and which velocity difference were used; the depth-averaged one depends on how far from the layer the measurement extended, and grows without limit as it extends further.

Five numbers, one name

The Richardson number has a threshold at a quarter, and the quarter belongs to one of the five quantities that go by the name. On a single tanh shear layer they run from J to seventeen J, and the largest of them grows without limit as the measurement is extended further from the layer.

turbulence · Stratification Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

A stratified sea blocks a current sooner and traps less

A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.

regimes · Geostrophy

Named alongside it

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

StratificationDimensionlessMixingModel limitRichardson numberThresholdConventionDispersion relationFroude numberGeostrophic balanceGroup velocityInternal wave

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