Hexagons remember how the heat was turned up

The threshold the walls decide computed the Rayleigh number at which a layer heated from below stops being still, and found it depends on nothing but what the top and bottom surfaces are allowed to do. Linear stability says that much and no more: it gives the threshold and the wavelength of the disturbance that grows first, and it says nothing about what the flow does once that disturbance is large. The truncation that cannot carry three times the heat went past the threshold, but it assumed the answer to the first question anyone asks about a convecting layer: what pattern it makes.

It assumed rolls. Bénard’s photographs of 1900, the first careful pictures of convection, showed hexagons.

One wavelength, two patterns #

Linear stability picks a wavelength but not a direction. At the threshold every roll whose wavevector has the critical length grows equally fast, whichever way it points, so the disturbance that emerges can be any sum of them. The two simplest sums are one set of rolls — parallel cylinders all turning the same way — and three sets of rolls at 120° to each other with equal strengths. Adding three cosine waves at those angles produces a field whose peaks lie on a triangular lattice, and the region closest to each peak is a hexagon: a cell with fluid rising in its centre and sinking at its edges, or the reverse.

So a hexagon is not a different kind of motion from a roll. It is three roll patterns superposed, and which of the two sums the layer chooses is a question about how the rolls interact once they are large enough to affect each other.

Three rolls and the term that joins them #

Just above onset the strengths of the three roll sets, $A_1$, $A_2$ and $A_3$, change slowly compared with anything else in the flow, and the equations they obey are almost entirely fixed by symmetry:

$$\frac{dA_1}{dt} = \varepsilon A_1 + g\,A_2 A_3 - A_1\left(A_1^2 + \lambda\left(A_2^2 + A_3^2\right)\right),$$

and the same with the indices turned round. Here $\varepsilon$ is how far above onset the layer is heated. The cubic terms say that every roll limits its own growth and, with a coefficient $\lambda$, the others’. The quadratic term is the interesting one. It joins three rolls whose wavevectors add to zero — exactly the three at 120° — and it lets two of them together feed the third.

That quadratic term is forbidden in a layer that looks the same upside down. Turning a symmetric layer over turns rising fluid into sinking fluid and changes the sign of every amplitude, and an equation that must be unchanged by that cannot contain a product of two amplitudes. The idealised layer of the Boussinesq equations is exactly that symmetric, and for it $g = 0$. A real layer is not, whenever the fluid’s properties change with temperature: a viscosity that falls as the fluid warms makes hot rising fluid move more freely than cold sinking fluid, and a cell with a hot centre is no longer the mirror image of one with a cold centre. A free surface on top breaks the symmetry more strongly still, which is why Bénard’s thin layers of spermaceti under air showed hexagons so clearly.

Why the angle is 120° #

The quadratic term is not available to any three rolls. It couples three sets only if their wavevectors add to zero, because a product of two waves is a wave whose wavevector is the sum of theirs, and it can feed a third only if that sum is the third’s wavevector reversed. All three must also have the critical length, since those are the only rolls growing. Three vectors of equal length that close into a triangle form an equilateral one, so the angles between the roll sets are exactly 120°.

That is why the pattern is hexagons and not squares. Two sets of rolls at right angles make a square lattice, but no third roll of the critical length closes a triangle with them, so no quadratic term joins them, and in this model a square pattern has only the cubic interactions that make one set suppress the other. Hexagons are the one arrangement the quadratic term can build, and the geometry of a triangle, not any property of the fluid, is what picks the angle.

The same counting says why the linear threshold could not have told the difference. At onset the amplitudes are infinitesimal, their products vanish faster than they do, and every direction and every combination of directions grows at the same rate. The pattern is decided at second order in the amplitude, where the triangle first matters, and the threshold at first order is blind to it.

Hexagons before onset #

The steady states of those equations can be written down. Rolls — one amplitude non-zero — have $A_1 = \sqrt{\varepsilon}$. Hexagons — all three equal — satisfy a quadratic, and because of the $g$ term its solutions exist slightly below onset, from $\varepsilon = -g^2/4(1+2\lambda)$. For $g = 0.1$ and $\lambda = 2$ that is five ten-thousandths below the threshold.

That is a subcritical bifurcation, and it means the linear threshold is not the whole story. Below onset the still layer is stable to small disturbances, as the linear calculation says; but a disturbance large enough, with the right three-fold shape, can reach the upper hexagon branch and convect there. Every mode decays and the flow grows anyway found a different route to the same lesson in pipe flow: a state stable to every infinitesimal disturbance is not therefore the state the system is in.

Stability separates the branches. Perturbing rolls with small amounts of the other two sets gives a pair of growth rates, $(1-\lambda)\varepsilon \pm g\sqrt{\varepsilon}$, so rolls are stable only once $(\lambda - 1)\varepsilon$ exceeds $g\sqrt\varepsilon$ — above $\varepsilon = g^2/(\lambda-1)^2$. Hexagons are stable up to $\varepsilon = g^2(2+\lambda)/(\lambda-1)^2$. For these coefficients the first is 0.0100 and the second 0.0400, and between them both patterns are stable.

The subcritical range adds a second loop to the one inside the window, below onset itself. A layer brought up past the threshold convects in hexagons; turned back down, it keeps them through the threshold and below it, to $\varepsilon = -g^2/4(1+2\lambda)$, because the upper hexagon branch is still there to sit on. Only below that does the convection stop and the layer fall still. Turned up again from rest, the still layer does not start convecting until the threshold proper, because nothing it contains is large enough to reach the branch.

For these coefficients that lower loop is five ten-thousandths of the heating wide, a sixtieth the width of the window above, and it grows as $g^2$ with it. It is small, and it is not nothing: a layer can be seen to convect at a heating where every linear calculation says it must be at rest, and the explanation is not an error in the calculation but a state it was never asked about.

Two starts, two endings #

The closed-form thresholds are checked by doing what a layer does: integrating the equations forward in time from a starting state and seeing where they come to rest. At ε = 0.025, in the middle of the window, a start with one roll set a little stronger than the others ends as pure rolls, the other two amplitudes decaying to nothing, at the closed form’s $\sqrt\varepsilon$. A start with all three nearly equal ends as hexagons, at the closed form’s amplitude to four figures.

Nothing distinguishes the two runs except where they began. Same fluid, same heating, same equations. Inside the window the final pattern is a record of the initial disturbance, and a layer in the laboratory, whose initial disturbance is whatever the apparatus happened to supply, can show either — which is part of why convection experiments were so hard to reproduce before this was understood.

A pattern that depends on the direction of travel #

An experiment rarely starts from a chosen disturbance. It turns the heating up slowly, or down, and watches. The figure does that: ε is ramped from just below onset to well past the hexagons’ threshold and back again, with a small random kick added to each amplitude at every step to stand in for the unavoidable noise of a real layer.

Going up, the layer starts convecting in hexagons — the quadratic term gives them a head start at onset — and keeps them through the whole window, until the hexagons lose stability and two of the three roll sets collapse. Coming down, the rolls are already there and stable, and they persist through the window too, until they in turn lose stability and hexagons return. At every heating inside the window, the pattern is the one the layer arrived in. That loop is the state a machine was started into in its simplest form, with the whole memory carried by three numbers.

Each switch happens a little beyond its threshold — at 0.0448 rather than 0.0400 going up, at 0.0085 rather than 0.0100 coming down. The lag is not an error in the thresholds. A state that has just lost its stability leaves it slowly, because its growth rate starts at zero, and the ramp keeps moving while the departure builds; a slower ramp brings both switches closer to the closed forms, and the direction of each lag is the direction of travel.

How fast the heat is turned decides where the switch is seen #

The lag between each switch and its closed-form threshold is a measurement of how slowly a state leaves once it has lost its stability, and it is worth putting a number on. The same ramp run four times faster — ε changing at two millionths per unit time rather than half a millionth — keeps hexagons going up until ε = 0.0513 rather than 0.0448, and keeps rolls coming down until 0.0070 rather than 0.0085. Both switches move away from their thresholds as the ramp speeds up, and both move towards them as it slows.

The limit of an infinitely slow ramp is the closed-form window, and a real experiment never reaches it. Near a threshold the growth rate of the departing state is proportional to the distance past the threshold, so the time to leave grows without limit as the threshold is approached, and a layer ramped at any finite rate always overshoots. The noise matters too: a larger random disturbance gives the departing state a bigger start and shortens the lag. The window measured in a laboratory is wider than the window in the equations, by an amount that depends on how patient the experimenter was and how quiet the apparatus.

The window closes as the square of the asymmetry #

All three thresholds are proportional to $g^2$, so the whole structure — the subcritical hexagons, the bistable window, the hysteresis loop — shrinks away together as the layer’s asymmetry is removed. Halving the asymmetry quarters the width of the window.

That makes hexagons a diagnostic. In a layer of a fluid whose properties barely change across the temperature difference, $g$ is tiny, the window is a sliver, and a layer heated even slightly past onset shows rolls. In a layer of oil with a viscosity that halves over a few tens of degrees, or with a free surface on top, $g$ is large enough that hexagons dominate a visible range of heating. The pattern reports on a symmetry the equations of the idealised layer assume and the real fluid does not have.

Without the asymmetry, rolls win #

Set $g$ to zero and repeat the run that ended in hexagons. The three amplitudes grow together at first, since the heating drives them equally. Then the cross-coupling takes over. With $\lambda > 1$ each roll set suppresses the others more strongly than it limits itself, so any small inequality grows: the strongest set gains at the expense of the other two, which decay, and the layer ends as pure rolls.

The value of λ is doing essential work there. If each roll set suppressed the others less than it limited itself — λ below one — the three amplitudes could grow together without any one winning, and a mixed pattern would be stable even in a symmetric layer. The equations here refuse that case outright, because buoyant convection between plates has λ above one and its rolls do compete; it is the competition, and not the quadratic term, that selects a single set when the symmetry is intact.

This is why the three numbers left of a fluid were roll amplitudes, and why the truncation and every essay built on it used rolls. A Boussinesq layer between two plates is exactly symmetric, and rolls are its only stable pattern near onset. The hexagons in the photographs belong to layers that were not Boussinesq — in Bénard’s case, mostly because the surface tension of the free surface varied with temperature and pulled the surface from warm regions towards cool ones, a surface stress driven by a gradient in tension that has nothing to do with buoyancy at all.

Not the pattern that carries most heat #

The excess heat a convecting layer carries is, to leading order, proportional to the sum of the squares of its roll amplitudes. For rolls that is ε; for hexagons, three times the square of the hexagon amplitude. Just above onset hexagons carry more — the quadratic term has pumped their amplitude up — but the two curves cross at ε = 0.0075, and past that rolls carry more heat at the same heating.

Inside the window, then, a layer in hexagons is carrying about four-fifths of the heat it would carry as rolls, and it stays in hexagons all the same until they become unstable. There is an old and attractive idea that a convecting layer arranges itself to carry heat as efficiently as possible — a principle of maximum heat transport, proposed for turbulent convection as a way of closing the problem without solving it. These equations do not obey it. They keep whatever stable state they are in, and the heat flux is a consequence of the pattern, not the thing that chooses it.

What the amplitude equations leave out #

Where the coefficients come from. The values of $g$ and $\lambda$ are not computed here. For a real fluid they follow from an expansion of the full equations about the threshold, with $g$ proportional to how strongly the viscosity and other properties vary across the layer; Busse carried out that calculation in 1967. The thresholds scale with them exactly as drawn, and their numbers depend on the fluid.

Space. The three amplitudes are uniform across the layer. A real layer has side walls, defects where patches of different orientation meet, and slow variations of the amplitude in space, which need gradient terms in the equations and bring instabilities of their own.

Other patterns. Squares, and rolls of a slightly different wavelength, compete as well. Rolls in particular are stable only inside a region of wavelength and heating — the Busse balloon — beyond which they break up by instabilities the three-amplitude model cannot represent.

Distance above onset. The equations are an expansion in small ε. Far above onset more modes matter, as the truncation’s heat flux showed for rolls, and the neat thresholds become numerical questions.

Bénard, Rayleigh, and the asymmetry #

Bénard photographed hexagonal cells in thin heated layers in 1900. Rayleigh’s analysis of 1916 explained the onset of convection from buoyancy and gave the cells’ size reasonably well, and for forty years the hexagons were taken as buoyancy’s pattern. Pearson showed in 1958 that in Bénard’s thin layers with a free surface the driving was mostly surface tension, not buoyancy; Busse and others in the 1960s showed why buoyant convection in a symmetric layer makes rolls, and why a broken symmetry makes hexagons. The first photographs of the subject showed the case that the first theory of it did not describe, and it took the question of symmetry to reconcile them.

Still open: the defects between the patches #

Every calculation here treats the layer as one uniform patch of pattern. A real large layer never is: patches of rolls with different orientations meet along boundaries, hexagonal lattices contain cells with five or seven sides, and those defects move, slowly, as the pattern relaxes. Their number and arrangement are not free, because a pattern of cells with stagnation points at their centres and corners is subject to the integer count a flow’s stagnation points cannot break, and a defect is where that count is being kept.

How fast such a pattern coarsens, and whether it ever reaches the perfect lattice the amplitude equations describe, is the question that joins pattern selection to the slow dynamics of the defects — and to how many things a flow must be told at the side walls, which set where the patches begin.