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

Two-dimensional turbulence — where it appears

Turbulence in a flow with no variation along one direction, where vortex stretching is absent and enstrophy is conserved alongside energy. The two conservation laws together send energy towards larger scales rather than smaller ones, which is the opposite of the three-dimensional cascade.

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

The condensate, and the limit of no friction which does not exist. In a steady state the friction must remove everything the forcing puts in, so the energy is eps/(2 alpha) and the coherent velocity is sqrt(eps/alpha) — exactly a minus one half power, checked to 10⁻¹². As the friction is weakened the condensate grows without bound: the limit alpha to zero is not a flow with a weak condensate, it is a flow with no steady state at all.

Where the inverse cascade stops

Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.

turbulence · Two-dimensional 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 A count of states that has a maximum in it. How many configurations of thirty point vortices have each energy, sampled from the measure their own Hamiltonian defines — which is the area measure, because a vortex's coordinates are its own conjugate pair. The count peaks at an energy of -0.054 and falls away on both sides, which no ordinary system's does. Above the peak, adding energy reduces the number of ways of arranging the fluid.

An equilibrium a three-dimensional flow cannot have

The condensate an inverse cascade ends in is treated as what is left over when the energy has nowhere further to go. It is not a remainder. A two-dimensional fluid's phase space is its own region and therefore has finite volume, so its entropy has a maximum, its temperature changes sign, and the clustered state above that point is an equilibrium.

turbulence · Two-dimensional

Named alongside it

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

Inverse cascadeModel limitCondensateVorticityAnisotropyBox scaleConservationDimensionlessDispersion relationEnergy fluxEnstrophyEntropy

All concepts