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 2 essays across one field — 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

Named alongside it

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

Dimensionless Model limit Anisotropy Buoyancy frequency Geophysical Group velocity Internal wave Inverse cascade Linear stability Mixing Regime Richardson number

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