Concept

Dispersion relation — where it appears

The relation between a wave's frequency and its wavenumber, which fixes the speed it travels at and whether that speed depends on the wavelength. A wave whose frequency falls with wavenumber is one that can only outrun a process at large scales.

Named by 9 essays across 5 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

The barrier is pinched to nothing in one direction. Where the inverse cascade is stopped, drawn in the plane of wavenumbers. The curve is the scale at which a Rossby wave oscillates as fast as an eddy turns over, and inside it the cascade cannot proceed. It is not a circle: it closes to a point on the axis of modes with no variation in longitude, so the cascade runs on unimpeded towards the largest scales in exactly one direction — and what it makes there is a band.

A cascade that arrives as stripes

Two-dimensional turbulence sends its energy upward in scale and, on a plane, piles it into a pair of vortices filling the box. On a rotating planet one extra term in the vorticity equation blocks that in every direction but one, and what arrives at the large scales is not a vortex at all but a set of bands.

turbulence · Two-dimensional

The drip spacing is set by the depth of the layer. The spacing of the fastest-growing wave, in units of 2π capillary lengths, against the depth of the hanging layer in capillary lengths, for water, glycerol and honey. A thin film of any of them drips at √2. Deep water drips at √3. Deep glycerol and honey keep growing past √3, their fastest wave as long as the layer is deep, because viscosity slows short waves more than long ones.

How far apart a ceiling drips

A layer of liquid hanging from a ceiling is heavy fluid over light, and every ripple on it longer than about seventeen millimetres grows. Which ripple grows fastest, and so how far apart the drips form, is usually given as one number. It is at least three, and what chooses between them is not the liquid's surface tension but the depth of the layer.

turbulence · Instability

The unburnt gas bends round a wrinkle, and the bending makes it grow. Streamlines of the unburnt gas flowing up towards a flame with a small sinusoidal wrinkle, one wavelength across, for a density ratio of seven, with the wrinkle exaggerated. The expansion behind the flame pushes back on the gas ahead of it where the flame bulges forward, so the streamlines spread there and the gas arrives slower, and they crowd together where the flame lags, so the gas arrives faster. A flame that burns into the gas at a fixed speed therefore advances further where it was already ahead.

A flat flame is unstable at every size

A flame expands the gas it burns, and the expansion pushes back on the fresh gas ahead. Where the flame bulges forward the fresh gas slows, so the bulge burns further forward; where it lags, the gas speeds up and it falls further back. Every wrinkle grows, the shorter ones faster, and what finally gives a real flame a size is how its burning speed responds to its own curvature.

kinematics · Dilatation

The slow bending wave, and where the cut-off stops describing it. The frequency of the slowest bending wave on a Rankine vortex — a helical wobble of the whole core that turns against the flow — against the wavenumber times the core radius, from Kelvin's exact relation and from his long-wave formula, which the cut-off method used for vortex pairs reproduces. They agree for long waves. Past ka = 1.44 the long-wave formula reverses sign and the exact wave does not, which is why a cut-off model's instabilities at such wavelengths are not real.

A vortex is a waveguide

A spinning core is stiff in a way still fluid is not, and it carries waves along its length: an infinite family of them for every pattern round its axis, travelling at up to 0.83 of the swirl speed at its edge. The slowest is a helical bend that turns against the flow, and it is the wave every model of a bending vortex has been borrowing without saying so.

inviscid · Swirl

Whether a sloping ceiling drips in place is one ray's growth. How fast a disturbance grows as seen by an observer moving along the ceiling at speed v, for four speeds at which the film carries its disturbances downhill, in the film's own units. On a flat ceiling (V = 0) the growth peaks at a quarter for the observer standing still and falls to zero for one running at ±1.622. Tilting the ceiling slides the whole curve downhill. While the observer at the point disturbed, v = 0, still sees growth, the ceiling drips where the disturbance began; at V = 1.622 that observer sees none, and above it the disturbance grows only while it is carried away.

A sloping ceiling drips downhill, or not at all

A film hanging from a level ceiling drips where its ripples form. Tilt the ceiling and the film flows downhill, carrying its ripples with it, and at some slope they are carried away faster than they spread and the ceiling stops dripping in place. That slope is set by one number, 1.622 — the speed at which a level ceiling's disturbance spreads — and for water it is tiny: a degree for a tenth of a millimetre of film, six for half a millimetre. Past it the ripples still grow, and still drip, but downhill, at a distance that grows with the slope: a ceiling shorter than that delivers its water to the edge.

turbulence · Instability

A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

kinematics · Dilatation

Viscosity slows every ripple, and the short ones most. The growth rate of a varicose ripple on a liquid jet against its wavenumber times the jet's radius, at Ohnesorge numbers of 0, 0.1, 1 and 10, in units of the capillary time. Every ripple longer than the circumference grows. Viscosity damps each one in proportion to the square of its wavenumber, so the short ones lose most: the fastest moves from ka = 0.697 inviscid to 0.344 at Oh = 1 and 0.123 at Oh = 10, and its rate falls from 0.343 to 0.0114.

Viscosity lets a jet break, later and into bigger drops

A thread of honey falls for metres before it breaks, and a thread of water for centimetres, which suggests that viscosity holds a jet together. It does not: it cannot stop any ripple longer than the jet's circumference from growing. It slows them, the short ones most, so the ripple that wins is longer, it takes the viscous time rather than the capillary one to win, and each drop it makes is bigger. The wavelength grows as the square root of the Ohnesorge number and the drop as its sixth root.

regimes · Atomisation $A spread of sizes smears the stop band and pushes the fast band up. The phase speed of sound against frequency in water carrying air bubbles at a void fraction of 10⁻³: all of one millimetre radius, and spread from 10 µm to 1 mm as n ∝ R^(−q) with q = 0, 3/2 and 10/3. Every cloud starts at Wood's speed, 312 m/s, because at low frequency every bubble is driven below its resonance. The single size jumps through its stop band near 3 to 13 kHz into a fast band. The distributions climb through a band as wide as their resonances, 3.2 kHz for the largest to 301 kHz for the smallest, and the more small bubbles they carry the later they emerge faster than water.$

The more small bubbles a cloud holds, the later it turns fast

Bubbles of one size give water three sound speeds: slow below their resonance, nothing in a stop band, faster than water above it. A cloud of many sizes has every resonance at once, so at any frequency its small bubbles soften it and its large ones stiffen it. The stop band becomes a wide band of attenuation, the fast band survives above it, and where the fast band starts is set by how much of the gas is in the smallest bubbles.

compressible · Speed of sound

Named alongside it

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

Model limit Linear stability Eigenvalue Group velocity Instability Surface tension Vorticity Buoyancy Capillary length Dimensionless Growth rate Lubrication film

All concepts