Decaying flows end on the sinh side

The streamfunction says which relaxed state solved the relaxed states of a two-dimensional flow in a periodic square for the three theories that predict them — Montgomery and Joyce’s point-vortex mean field with its $\zeta = \sinh\psi$, the Miller–Robert–Sommeria maximum-entropy theory whose two-level member is a tanh, and selective decay’s straight line — and looked for one number that told them apart in a real flow’s data. It found one. The flatness of the streamfunction, $\langle\psi^4\rangle/\langle\psi^2\rangle^2$, is 9/4 for the linear dipole, above it for every sinh dipole and below it for every tanh dipole, and it barely moves when small-scale noise is added that would wreck the vorticity’s statistics.

It ended with the obvious next step and did not take it. The rule is a prediction about data the calculation had not touched. Run a decaying flow, from several kinds of start, follow the streamfunction’s flatness as it decays, and see which side of 9/4 it settles on — and whether the side depends on how the flow began. That is what the maximum-entropy theory says should happen, since it remembers the initial distribution of vorticity, and what the mean-field theory says should not.

This essay runs the decay. The answer is one-sided, and more interesting than either theory’s claim to the result.

The runs #

The vorticity equation is solved on the doubly periodic square of side $2\pi$, pseudo-spectrally on a $64^2$ grid with the two-thirds rule, its nonlinear term by fourth-order Runge–Kutta and a hyperviscous term, $\nu(-\nabla^2)^4$, integrated exactly. Hyperviscosity is the standard compromise for this question: it removes the enstrophy that cascades to the grid while leaving the energy, which the relaxed theories conserve, almost untouched — the runs here lose between one and seventeen per cent of their energy, nearly all of it in the first few time units, while their enstrophy falls tenfold. It is not the Navier–Stokes equation, and that is stated rather than hidden.

Three families of initial vorticity, two seeds each, every one normalised to the same energy:

• random phases on a spectrum peaked at wavenumber six — the usual start of a decaying simulation;
• two-level patches, two dozen discs of vorticity $+1$ and $-1$ — the start the tanh theory is built on, since it assumes the vorticity takes two values and remembers the area of each;
• a $k^{-3}$ spectrum from wavenumber three to twenty, the shape an enstrophy cascade leaves.

Each is run to $t = 600$, in units of the initial velocity over the box. The first essay on decaying two-dimensional turbulence, the cascade that runs backwards, described what happens along the way: like-signed vortices merge, filaments are shed and dissipated, and the energy collects in ever larger structures.

Every run crosses 9/4 and stays above it #

All three start near Gaussian, a streamfunction flatness near three for a random field, and dip towards 9/4 as the small scales are stripped away and the field is dominated by its largest modes. Then the mergers concentrate the vorticity into a few strong vortices, and the flatness rises again, above 9/4, and settles. At $t = 600$ it is 4.10 for random phases, 2.92 for two-level patches and 3.49 for the $k^{-3}$ start. The second seeds land at 3.94, 2.97 and 3.48. No run ends below 9/4. By the rule the previous essay derived, every one of them is on the sinh side — including both runs that began as two levels of vorticity.

On the branch, not merely above the line #

A side of a line is a weak test. A stronger one asks whether the decayed states are sinh dipoles, not merely on the sinh side. Each relaxed theory is a curve in the plane of streamfunction flatness against $Z/E$, the ratio of enstrophy to energy, because both are independent of the scale of $\psi$: the tanh branch falls below 9/4 and stays near $Z/E = 1$, and the sinh branch climbs steadily. A decayed state is a point, and the test is whether the point lies on a curve.

Four of the six do. The two runs begun as patches sit within 0.005 and 0.055 of the sinh branch at their own $Z/E$, about 1.38. The two begun on a $k^{-3}$ spectrum sit within 0.04 and 0.07, at $Z/E$ near 2.1. The two random-phase runs sit above the branch, by about 0.30, at $Z/E$ of 2.6 and 3.1 — on the sinh side of 9/4 by a wide margin, but not on any curve the three theories draw.

The vortices’ concentration, a second statistic #

The vorticity’s own flatness gives a second, independent reading. A tanh dipole’s vorticity is flat-topped, with a flatness below two; a sinh dipole’s is peaked, and its flatness climbs steeply with $Z/E$ — 10 at $Z/E$ of 1.38, 27 at 2.2, 48 at 3.0. The decayed states’ flatnesses run from 8 to 30, which rules out the tanh branch at once. As ratios to the sinh branch at the same $Z/E$, the patches’ runs are 1.04 and 0.77 and the $k^{-3}$ runs 1.10 and 0.96: sinh dipoles in this statistic too, within the scatter of single runs. The random-phase runs are 0.60 and 0.83 — strongly concentrated vortices, but less concentrated than the sinh dipole carrying their enstrophy.

The two statistics agree on what the random-phase runs are. Their streamfunction is more spread, and their vorticity less peaked, than the sinh dipole at their $Z/E$. Both are what a state with more than one vortex of each sign would show: several concentrated vortices that have not finished merging spread the streamfunction across the box and share the enstrophy among more peaks. At $t = 600$ the random-phase runs are still slowly merging; the others have reached their pair.

The relation itself #

The statistics are summaries. The relation they summarise can be looked at directly: a relaxed state has $\zeta = F(\psi)$, so its grid points plotted as vorticity against streamfunction fall on one curve, and the curve’s shape is the theory. For the decayed state that began as two-level patches the points do fall on one curve, tightly, and it bends away from the origin — convex, the shape of a sinh. The best sinh relation fits it with an rms error of 0.14 of the rms vorticity. The best tanh relation can do no better than 0.48, which is exactly what a straight line achieves: a tanh bends towards the axis, and fitted to a curve that bends away from it, the best it can do is make its bend vanish.

That is the most direct statement of the result. The flow that began as two levels of vorticity, the tanh theory’s own case, has forgotten that it had two levels. Its final relation is the mean field’s.

Six runs, one verdict #

Across all six runs the verdict is the same. The sinh fits leave 0.14 to 0.34 of the rms vorticity; the tanh fits 0.46 to 0.73, in every case equal to the straight line’s. The random-phase runs fit worst, consistent with their not having finished — a state with several vortices of each sign is not a single function of $\psi$, and no relation fits it perfectly. But even they are much nearer a sinh than a tanh.

Why the tanh loses is the point worth taking from this. The maximum-entropy theory’s distinctive claim is that the flow remembers how much area each value of vorticity occupies, because an ideal two-dimensional flow only rearranges vorticity. A flow with any dissipation at small scales does not conserve those areas. Every filament a merger sheds is stretched to the grid scale and removed, and with it goes vorticity at intermediate levels; the area at each level is precisely what the cascade destroys. The point-vortex mean field never claimed to remember it. What survives the cascade is the energy, which hyperviscosity leaves, and the strong cores of the vortices, which the cascade cannot reach — and a gas of strong, nearly point-like vortices with conserved energy is the sinh theory’s subject. What decay never forgets is about large scales in three dimensions, but the logic is the same: an invariant survives the decay only if the decay cannot reach the scales that carry it.

Watching the two levels go #

The loss of the two levels can be followed in the runs themselves. At the start the patch field’s vorticity takes the values $+1$, $-1$ and zero almost everywhere, with a thin smoothed rim round each disc; its vorticity flatness is 3.7, the value for a field that is mostly zero with flat-topped islands. Within the first fifty time units the patches of each sign find one another — vortices move each other, and two of one sign orbit and, when close enough, merge. Each merger wraps the two cores round one another and throws out long arms of vorticity at the original levels, and those arms are sheared thin by the merged vortex’s own rotation, as a vortex pair carries eddies no snapshot sees traced for the fluid around a pair. Within a few turnovers the arms are at the grid scale and the hyperviscosity removes them.

What is left after the mergers is not two flat-topped patches. It is two peaked vortices whose cores are the overlapped remains of many original discs, where vorticity of one sign has been piled up, and whose skirts are the smoothed residue of everything sheared off. By $t = 50$ the vorticity flatness has risen from 3.7 to 11 and $Z/E$ has fallen from 6.5 to 1.4; from then on the run barely changes. A peaked vortex surrounded by weak vorticity is what a cloud of like-signed point vortices looks like when averaged — the sinh theory’s picture — and nothing in it remembers that the vorticity once took only two values.

What a point vortex is not warned against reading a smooth vortex as a point. Here the dynamics has done the reverse: it has made the smooth field into something the point-vortex statistics describe.

How far the decay relaxes #

None of the runs reaches the linear dipole, whose $Z/E$ is one. The ratio falls fastest at first, from 71 for random phases and 30 for the $k^{-3}$ start, and by $t = 600$ it is 3.1, 1.4 and 2.2 and still falling slowly. That is itself a result about the theories. Selective decay predicts the minimum-enstrophy state, the linear dipole at $Z/E = 1$, and a decaying flow approaches it only as its vortices smear; with hyperviscosity, which cannot touch scales the vortices no longer have, the decay stops short, at the strongly peaked end of the sinh family. The flow relaxes to a maximum-entropy state of the vortices it has, not to a minimum-enstrophy state of the field.

Checks on the solver #

The two-dimensional transform is figure-kit’s, applied along rows and columns; its round trip on a random field is exact to $10^{-16}$ and it satisfies Parseval’s identity exactly. The field $\sin x + \sin y$ is a steady solution of the Euler equation, its vorticity a linear function of its streamfunction, and the solver’s nonlinear term on it is $2\cdot10^{-13}$; its streamfunction flatness is 9/4 to rounding, the number the whole comparison turns on. With the hyperviscosity switched off, a random field’s energy and enstrophy are conserved over four time units to two parts in ten million, the time-stepping’s error. The tests also refuse a grid that is not a power of two, an end time of zero and an initial condition the solver does not define.

What a 64-point box cannot say #

Resolution. Sixty-four points across the box resolve the final vortices well and the mergers that make them only adequately. A finer grid with a smaller hyperviscosity would leave stronger, smaller cores and could move the final $Z/E$; whether it moves the side of 9/4 is the next thing to check, and the prediction from the argument above is that it does not.

Hyperviscosity. It is a model of a dissipation, chosen to keep the energy. With ordinary viscosity the energy decays too, and the final states drift slowly towards the linear dipole as the vortices diffuse.

Six runs. Two seeds of three families is a sample, not a statistical survey. The verdict is unanimous on the side of 9/4 and on the fits, which is what makes six runs informative, but the random-phase runs’ departure from the branch is measured on two.

One box. A doubly periodic square has the dipole as its relaxed pair, and where the inverse cascade stops is set by the box. Other domains have other end states, and in a rectangle or on a rotating plane the stripes compete.

What the sign rule is good for #

The rule has now been tested on data it was not built from, and it did what was claimed of it: the side of 9/4 was decided early, stayed decided, and agreed with the direct fit of the relation. That makes it usable where the direct fit is not. A laboratory flow — a soap film, a thin layer of salt water driven by magnets, a rotating tank — gives a velocity field on a window, noisy at small scales. The vorticity is the derivative of that field and inherits its noise; the streamfunction is its integral and averages the noise away. A flatness computed from the streamfunction over the window can be compared with 9/4 where a scatter plot of vorticity against streamfunction would be a cloud.

What the rule cannot do is decide the theory from one sign. Four of the six runs here are sinh dipoles; two are something sinh-like but unfinished, and the sign alone puts them in the same class. The branch test — the point against the curve at its own $Z/E$ — is what separates them, and it needs $Z/E$, which needs the vorticity again. An equilibrium a three-dimensional flow cannot have explained why a two-dimensional flow has relaxed states to test at all; the sign tells which family a measured one belongs to, and the branch whether it has arrived.

The convention: flatness and Z/E #

The streamfunction’s flatness is $\langle\psi^4\rangle/\langle\psi^2\rangle^2$ over the box, with the mean of $\psi$ zero; the vorticity’s likewise. Both are independent of the scale of the field, as is $Z/E$, the enstrophy over the energy, so every relaxed theory is a single curve in their plane and a decayed state a point. Time is in units of the initial velocity over the box side. The vorticity is $\zeta = -\nabla^2\psi$ and the velocity $(\partial_y\psi, -\partial_x\psi)$.

Who ran it first #

Matthaeus, Stribling, Martinez, Oughton and Montgomery found decaying two-dimensional turbulence relaxing towards sinh-Poisson states in high-resolution simulations in 1991, and Montgomery and colleagues followed it up; Brands, Maassen and Clercx in 1999 compared decaying runs with both the mean-field and the Miller–Robert–Sommeria predictions and found the outcome sensitive to how the flow was dissipated. The streamfunction-flatness test is the previous essay’s; its application to decaying runs from two-level starts, and the observation that the tanh fit collapses onto the straight line, are what this calculation adds.

Still open: whether resolution moves the verdict #

The six runs here share one grid and one hyperviscosity. The next calculation repeats the two-level-patch runs at 128 and 256 points with hyperviscosity scaled down to keep the same ratio at the grid, and with ordinary viscosity at a matched Reynolds number, and asks whether the streamfunction’s flatness at the end stays above 9/4 — or whether, with less of the cascade removed, enough of the initial two levels survives in the vortex cores for the tanh theory to claim the final state. If the side changes with resolution, the sign rule measures the dissipation as much as the dynamics, which would be worth knowing before applying it to a laboratory soap film.