Two-dimensional — the series · Fluid Flow & Aerodynamics

80 per cent of the energy goes to the larger scale. One unit of energy is taken out of the middle wavenumber and shared between its two neighbours. Two conservation laws decide the split completely: the energy must add up, and so must the enstrophy, which weights each wavenumber by k². For (1, 2, 4) the answer is that 80.0 per cent of the energy goes up in scale and 80.0 per cent of the enstrophy goes down. There is no model of turbulence anywhere in that: it is two linear equations in two unknowns, and it is why a two-dimensional flow organises itself into large vortices while its gradients get finer.

The cascade that runs backwards

Three-dimensional turbulence carries energy from large scales to small ones and dissipates it. Take away one dimension and the term that does it vanishes identically, a second quantity becomes conserved, and two conservation laws between them force the energy to go the other way — up in scale, into ever larger vortices.

part 1 · turbulence

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.

part 2 · turbulence

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.

part 3 · turbulence

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.

part 4 · turbulence

Three relaxed states with the same energy and enstrophy. The vorticity along the diagonal of the periodic square, through the centres of both vortices of the dipole, scaled by its rms value, for the three relations at the same ratio of enstrophy to energy, Z/E = 1.1 — except the linear state, which exists only at Z/E = 1. The sinh state concentrates its vorticity into sharp cores; the tanh state spreads it into flat-topped patches with steep edges; the linear state is a sine. All three carry the same two quadratic invariants in proportion.

The streamfunction says which relaxed state

Decaying two-dimensional turbulence ends in a large pair of vortices, and three theories say what that pair should look like: a sinh relation between vorticity and streamfunction, a tanh, or a straight line. Their scatter plots differ only in curvature, and a real flow's scatter hides curvature. Solve the three states in the same periodic box, at the same energy and enstrophy, and a statistic that separates them turns out to be one nobody looks at: the flatness of the streamfunction, which sits above the straight line's value for every sinh state and below it for every tanh state, and does not move when unrelaxed small eddies are added.

part 5 · turbulence

Four of six decayed states sit on the sinh branch; the other two lie above it. The streamfunction's flatness against the enstrophy-to-energy ratio Z/E: the sinh and tanh relaxed dipoles as curves, from the linear dipole at Z/E = 1 outwards, and the six decayed states at t = 600 as points. The tanh branch falls below 9/4 and stays near Z/E of one; the sinh branch rises. The two runs begun as two-level patches lie on the sinh branch at their own Z/E, 0.0049 to 0.055 from it, and so do the two begun on a k⁻³ spectrum, 0.04 to 0.072. The two random-phase runs lie above it, by 0.3 to 0.31: on the sinh side of 9/4, but not sinh dipoles.

Decaying flows end on the sinh side

Three theories predict the state a decaying two-dimensional flow relaxes to, and the streamfunction's flatness tells them apart: above 9/4 for the point-vortex theory's sinh, below for the two-level theory's tanh. Run the decay from three kinds of start and every run ends above 9/4 — including the one begun as two-level patches, the case the tanh theory was built for. Four of six land on the sinh branch itself; the other two are sinh-like without being sinh.

part 6 · turbulence