Concept

Negative temperature — where it appears

A statistical state in which the most energetic configurations are the most probable, possible only for a system whose energy is bounded above. Point vortices in a bounded domain have such states, and in them like-signed vortices cluster into large structures.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

turbulence · Two-dimensional

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.

turbulence · Two-dimensional

Named alongside it

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

Enstrophy Flatness Inverse cascade Model limit Statistical mechanics Streamfunction Two-dimensional turbulence Vortex patch Vorticity

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