The threshold the walls decide

A threshold with a closed form computes the onset of convection in a layer heated from below and finds the answer that can be written down: with both surfaces free, the critical Rayleigh number is $27\pi^4/4 = 657.5114$ at a wavenumber of $\pi/\sqrt2$, exactly. It also records what it could not do — the rigid case, which has no closed form and whose value of 1707.762 that essay quotes rather than produces.

This one produces it, and finds that the interesting quantity is not the number but the ratio between the three cases.

The problem, and why it is an eigenvalue #

Linearise the Boussinesq equations about the conducting state and look for the steady marginal solution — exchange of stabilities holds here, which is a theorem rather than a convenience — and the vertical velocity and the temperature perturbation satisfy

$$(D^2 - a^2)^2 W = \mathrm{Ra}\,a^2\Theta, \qquad (D^2 - a^2)\Theta = -W,$$

with $a$ the horizontal wavenumber. Combining gives a sixth-order equation,

$$(D^2 - a^2)^3 W = -\mathrm{Ra}\,a^2 W,$$

whose solutions exist only for particular values of $\mathrm{Ra}$ at each $a$. Those are the marginal Rayleigh numbers, they form a discrete increasing sequence, and the critical value is the minimum over $a$ of the smallest of them.

Nothing in that is a comparison of two terms. The Rayleigh number is a term ratio — buoyancy against the two diffusions that fight it — and it is one where they balance, and at that value the operator has no non-trivial solution, so nothing happens. The threshold is where an operator acquires one, and the value is whatever the operator says.

Six conditions, and where each of them lives #

A sixth-order problem needs six boundary conditions, and getting them into the right places is what makes the calculation honest rather than merely convergent.

Each surface supplies three. $W = 0$, because nothing crosses it. Then either $DW = 0$ for a rigid wall — no slip — or $D^2W = 0$ for a free surface, which is no shear. And $\Theta = 0$ for a conducting surface held at a fixed temperature.

The trick that keeps them separate is to build two operators rather than one. $\Theta$ carries its own condition inside the operator $L = D^2 - a^2$, and $W$ carries its two inside the biharmonic $B = (D^2-a^2)^2$, and the equation solved is $LBW = -\mathrm{Ra}\,a^2 W$. Six conditions in the two places they belong, rather than smeared over one stencil.

The entire difference between 657 and 1708 is one sign in the ghost value at the wall: the value just outside the domain is $+W_1$ for a rigid surface, from $DW = 0$, and $-W_1$ for a free one, from $D^2W = 0$. One character in the discretisation, a factor of 2.6 in the answer.

Checking it against the one case that can be checked #

An eigenvalue quoted to seven figures off a hundred-point grid needs evidence, and the free–free case supplies it: it has a closed form at every wavenumber, not only at the minimum, so the solver’s error can be measured rather than estimated.

$\mathrm{Ra}(a) = (\pi^2 + a^2)^3/a^2$, and comparing at three wavenumbers — not one, since agreeing at the minimum alone would agree with a curve of the wrong shape — gives errors of a few parts in a hundred thousand after Richardson extrapolation, with an observed convergence order between 1.5 and 2.2 against the 2 a second-order scheme must give.

Then the search: golden section over the wavenumber gives $a_c = 2.2196$ against the exact $\pi/\sqrt2 = 2.22144$, and $\mathrm{Ra}_c = 657.408$ against $657.5114$ — sixteen parts in a hundred thousand.

With that established, the other two cases are computed the same way and compared with the published values: 1100.697 against Pellew and Southwell’s 1100.65, and 1707.757 against Jeffreys’ 1707.762.

What the boundary condition is doing physically #

The factor of 2.6 has a mechanism, and the eigenfunctions show it.

Convection has to move fluid horizontally as well as vertically: a cell is a circulation, and the returning flow at the top and bottom of it runs along the boundaries. A free surface lets that happen at no cost. A rigid wall forbids it, so the horizontal motion has to take place a little way inside the layer, and the shear needed to accomplish it is dissipated viscously.

More dissipation means more buoyancy is needed to sustain the motion, which means a higher Rayleigh number. Two rigid walls cost the most, one rigid wall costs an intermediate amount, and the ordering is forced.

The cell shape changes with it. The critical wavenumber rises from 2.2196 to 3.1158, so the cell narrows from 2.83 layer-depths across to 2.02 — very nearly square. The fluid narrows its cells to keep the overturning away from the expensive boundaries, and pays a different price in doing so, and the eigenvalue is the compromise.

The numbers, and how far apart they are #

Three rows, one fluid, one depth, one temperature difference. The spread in the threshold is a factor of 2.60 and the spread in the cell width is a factor of 1.40, and nothing has changed but a boundary condition at each surface.

It is worth putting that beside the precision anybody has in the inputs. A convection estimate for a layer of oil, a layer of magma or the air gap in a window needs the thermal expansion coefficient, the viscosity, the diffusivity, the depth and the temperature difference, and it is doing very well if the worst of those is known to twenty per cent — which enters the Rayleigh number linearly or as a cube. The boundary-condition factor is comparable with all of the property uncertainties together, and unlike them it is a choice rather than a measurement.

Which case a real experiment is #

The three are not equally realistic, and the useful one is the one with the largest number.

Two rigid walls is a laboratory Rayleigh–Bénard cell: fluid between two metal plates, and its convection is what a millionth of a degree is enough to start. This is what every quantitative experiment on convective onset has been, and 1707.76 is the number measured — to about a per cent, by many groups, since Schmidt and Milverton in 1935.

One rigid, one free is a pan of water on a stove, or a layer of oil with air above it. The threshold is 1100.65, and the free surface is only free if surface tension is negligible; if it is not, a completely different instability takes over — Marangoni convection, driven by tension gradients rather than by buoyancy — and the two compete.

Two free surfaces is not a laboratory configuration at all. It is a mathematical convenience whose only virtue is that it can be solved in closed form, and its value is quoted constantly because of that.

So the number with a closed form is the one that describes nothing, and the two that describe something have to be computed. That is a common pattern and it is worth noticing when it happens.

And the walls at the side, of which this calculation has none #

The essay has argued that the horizontal boundaries decide the answer, and the calculation it argues from has no vertical ones. The layer is infinite, the disturbance is a single horizontal wavenumber extending forever, and the eigenvalue is a minimum over a continuum of wavenumbers. Every experiment ever performed has been in a box.

Sidewalls do two things, and the second is more serious than the first.

They raise the threshold. A lateral wall imposes no slip on the horizontal return flow of the cells nearest it, which is more dissipation for the same buoyancy, so a container of modest aspect ratio convects at a Rayleigh number appreciably above 1707.76 and approaches it only as the width grows to ten depths or more. They also quantise the wavenumber: the layer can no longer choose the optimum $a$ freely but must fit a whole number of cells across the box, so the minimisation that produced the 27/4 above is taken over a discrete set rather than a continuum, and lands on whichever member is nearest.

The second effect removes the sharpness. A sidewall is a place where the horizontal temperature is not uniform — the wall conducts, its material is not the fluid, and the isotherms bend where they meet it — and a horizontal temperature gradient drives a flow at any Rayleigh number whatever, however small. So in a real container the conducting state does not exist. There is always a slow circulation near the walls, growing steadily as the heating is increased, and what happens at 1708 is that the response steepens rather than that something begins.

A perfect bifurcation has been replaced by an imperfect one, in the technical sense: the pitchfork has been unfolded by a small forcing, its sharp corner has become a smooth curve, and the threshold is now an inflection rather than an event. Measuring 1707.76 to three figures therefore requires a container wide enough for the sidewalls to be irrelevant over most of it, sidewalls whose conductivity is matched to the fluid so that the corner forcing is small, and a criterion for reading a threshold off a smooth curve. The classic experiments that agree with the number to a per cent were built around exactly those three requirements.

That is worth setting against the claim two sections above that an eigenvalue threshold has no tolerance in it. Inside the idealisation, it has none: the operator either has a solution or it does not. The tolerance re-enters with the container, through the size of the imperfection, and it is a tolerance on the experiment rather than on the theory.

Which is the honest general form of this essay’s argument. The horizontal boundaries change the threshold by a factor of 2.6 and are usually chosen without thought. The lateral boundaries change whether a threshold exists at all as a sharp thing, and are usually not mentioned. In a problem whose answer is an eigenvalue, every boundary is a parameter — and the ones that are hardest to see are the ones the idealisation removed before anybody started computing.

Why the threshold is nowhere near one #

The general point this belongs to is that an eigenvalue threshold has no relation to the value at which the group’s terms balance.

The Rayleigh number is $g\alpha\Delta T d^3/\nu\kappa$, which is a perfectly ordinary term ratio: buoyancy over the product of the two diffusivities that oppose it. At $\mathrm{Ra} = 1$ those are comparable and the layer sits still. At 100 it sits still. At 1707.76 it does not, and the number is large for a reason that is entirely about the geometry of the operator.

Two factors of that size are visible in the eigenvalue. $\pi^4 \approx 97$ comes from the vertical structure: the mode must vanish at both surfaces, so it has at least a half-wavelength across the layer and each of the four derivatives contributes a $\pi/d$. And the minimisation over $a$ contributes 27/4, since the optimum wavenumber is a compromise between a cell too wide to carry heat efficiently and one too narrow to avoid conducting it away sideways.

$97 \times 6.75 = 657$. The whole of the free–free eigenvalue, in two factors, neither of which has anything to do with the balance of buoyancy against diffusion.

An eigenvalue is a different kind of answer #

It is worth being explicit about what changes when a threshold is an eigenvalue rather than a term ratio, because two things follow that do not follow for the other kind.

There is no tolerance. Asking for the Rayleigh number at which convection is “one per cent established” is not a question — the conducting state is stable below the threshold and unstable above it, and the amplitude of what grows is set by the nonlinearity rather than by the linear problem. The threshold does not move when the tolerance does, which is the test that separates the four kinds.

And it is exact. 1707.762 is not an engineering estimate; it is a number with as many digits as anybody cares to compute, defined by an operator, and a better experiment does not refine it — a better experiment tests whether the operator is the right one.

That combination is rare in fluid mechanics and it is why this problem has had the attention it has. It is one of very few places in the subject where a prediction and a measurement can be compared to three figures with nothing fitted in between, and where the agreement is therefore a real test of the equations rather than of a correlation.

What the picture cannot show #

Linear stability, and only that. Everything here answers the question of when the conducting state stops being stable to infinitesimal disturbances. It says nothing about what the flow does afterwards, which pattern is selected, or whether a finite-amplitude disturbance could trigger convection below the threshold — and for Rayleigh–Bénard with these boundary conditions it could not, since the bifurcation is supercritical, but that is a separate result.

Boussinesq. Density variation is kept only in the buoyancy term, which requires the temperature difference to be small compared with the absolute temperature. Real experiments with large temperature differences show non-Boussinesq effects that break the symmetry between top and bottom.

Conducting surfaces. A perfectly insulating boundary gives a different problem — the same sensitivity to what a wall is told that appears wherever a condition is replaced by an averaged one — with a critical Rayleigh number of 720 for free–free and a critical wavenumber of zero — the preferred cell is infinitely wide, which is a qualitatively different answer and depends on nothing but the thermal condition.

And the fluid has no Prandtl number in it, which is unusual for a thermal problem — the ratio of the two layers governs almost everything else in the subject. Exchange of stabilities makes the marginal state independent of the Prandtl number, which is why the threshold is one number rather than a family. That independence is exactly what fails for a rotating layer or a magnetic one, where oscillatory onset becomes possible and the Prandtl number enters the threshold.

Who found it, and when #

Bénard did the experiments in 1900 and Rayleigh the free–free theory in 1916, and the two have been attached to each other ever since despite Bénard’s own experiments having been dominated by surface tension rather than by buoyancy — his layers were thin and open-topped, which is the Marangoni case.

Jeffreys computed the rigid–rigid eigenvalue in 1928 by a method that took him a great deal of effort; Pellew and Southwell gave the definitive treatment in 1940, including the proof that exchange of stabilities holds. Chandrasekhar’s 1961 book is where most people meet the numbers, and it is where the ones quoted in this essay’s comparison come from.

The surprising connection is with what the number is used for. The critical Rayleigh number is quoted in textbooks as the boundary between conduction and convection in everything from double glazing to the Earth’s mantle, and it is quoted as 1708 — the rigid–rigid value — in cases where neither boundary is rigid. A magma layer beneath a crust has one rigid boundary and one that is not, and the mantle has neither. The factor of 2.6 between the cases is larger than the uncertainty in almost any of the physical properties those estimates use, and it is the part that gets copied without being chosen.

Where the ladder goes next #

Above this rung is the nonlinear problem: which pattern is selected, why hexagons appear when the fluid properties vary with temperature and rolls when they do not, and how the Nusselt number grows above the threshold — which is a millionth is enough and beyond it a numerical subject.

Beside it sits the closed form, which computes the case this one checks against, and the three numbers left of a fluid, which is what dimensional analysis leaves of the problem before any of it is solved. Below it is how many things a flow must be told, which is where the factor of 2.6 actually lives.