Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Stratification — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · turbulence
  2. 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.

    part 2 · turbulence
  3. Instability up to a quarter, and none past it. The fastest growth rate of a stratified shear layer against its Richardson number, on a profile whose gradient Richardson number is the same at every height. It falls smoothly towards zero and reaches it at a quarter: at Ri = 0.2499 the fastest mode still grows at 0.00106, and at 0.26 the solver finds no unstable mode at all.

    Sufficient, and not necessary

    A stratified shear layer whose Richardson number exceeds a quarter everywhere cannot go unstable. That is a theorem with an exact number in it. What it does not say — and what it is constantly read as saying — is that a layer below a quarter will.

    part 3 · turbulence

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