A threshold with a closed form

Everything else in this field has a threshold that is not quite a threshold. Pipe transition has no linear threshold at all. Flat-plate transition has one, three decades below where transition is observed. Shear layers have a criterion that is necessary and not sufficient.

Convection is the exception, and that is why it is worth a ladder of its own. A layer of fluid heated from below is stable below a definite value of a dimensionless number and unstable above it; the value is the minimum of an eigenvalue curve; and for one choice of boundary conditions that minimum has a closed form.

Why nothing happens at first #

The intuition that a fluid heated from below must immediately overturn is reasonable and wrong, and the reason it is wrong is a competition between three rates.

A parcel of fluid near the bottom is warmer than its surroundings, therefore less dense, therefore buoyant. If it rises, it enters cooler surroundings and its buoyancy relative to them grows, which is the destabilising feedback.

Two things oppose it while it travels.

Heat diffuses out of it. The parcel is warmer than what it is passing through, so it loses its temperature excess by conduction at a rate set by the thermal diffusivity κ. Lose enough of it and the buoyancy that was driving the motion is gone.

Momentum diffuses out of it. The parcel is moving relative to its surroundings, so it is dragged by viscosity at a rate set by ν.

The layer convects when the buoyant driving beats both diffusions over the time it takes to cross the layer. Collecting the quantities gives one dimensionless group:

$$Ra = \frac{g\alpha \Delta T\, d^3}{\nu\kappa}$$

with d the depth, ΔT the temperature difference, α the expansion coefficient and g gravity. Below some critical value of Ra, nothing moves.

The eigenvalue problem, and the curve #

The linearised equations for a disturbance of horizontal wavenumber a, with stress-free boundaries at top and bottom, reduce to a single condition:

$$Ra(a) = \frac{(\pi^2 + a^2)^3}{a^2}$$

This is a neutral curve: for each horizontal wavenumber, the Rayleigh number at which that disturbance neither grows nor decays. Above the curve the disturbance grows; below it, it decays.

The layer as a whole is unstable as soon as any disturbance can grow, so the threshold is the minimum of that curve over a — which is what makes the onset an optimisation rather than an evaluation.

Differentiating gives 2(π² + a²)² (2a² − π²)/a³ = 0, so a² = π²/2 and

$$a_c = \frac{\pi}{\sqrt 2} = 2.221441\ldots, \qquad Ra_c = \frac{27\pi^4}{4} = 657.5113\ldots$$

Found rather than quoted #

The site does not use the algebra above to produce the number. It scans the curve, brackets the minimum, and runs two hundred golden-section iterations on it, and then asserts that what it found agrees with the closed form to one part in 10⁶ — an assertion that fails the build if it does not.

That is a deliberate discipline and not a display. A neutral curve with a minimum in it is a shape that looks right whatever function produced it: an exponent mistyped, a π lost, a factor of two in the wrong place, and the figure would still show a U with a distinct bottom, and the essay would still read correctly. The closed form is the check and the search is the answer, in that order.

Measured: a_c = 2.221441 and Ra_c = 657.511364, against π/√2 = 2.221441 and 27π⁴/4 = 657.511364.

What the number is made of #

The Rayleigh number is worth taking apart, because its structure explains why convection is so easy to start in the atmosphere and so hard to start in a teacup.

The depth enters as d³. That is a very strong dependence: doubling the depth multiplies the Rayleigh number by eight, and a layer ten times deeper is a thousand times closer to convecting at the same temperature difference.

The two diffusivities enter as a product in the denominator, which is the mathematical form of the statement that both of them have to be beaten. A fluid with a large ν and a tiny κ is as hard to convect as one with the reverse.

Putting numbers to it: a 1 mm layer of water with a 1 K difference across it has a Rayleigh number of about 3, well below threshold and perfectly still. A 1 cm layer with the same difference is at about 3,000, comfortably above. The Earth’s atmospheric boundary layer on a sunny afternoon, a kilometre deep, is at something like 10¹⁷, and the Earth’s mantle — extraordinarily viscous, and three thousand kilometres deep — is at about 10⁷.

The d³ is why. It is also why the threshold is almost never the interesting question for a geophysical flow: everything at that scale is many orders of magnitude above onset, and the useful questions are about transport rather than about stability.

What the wavenumber predicts, and it can be checked in a kitchen #

The critical wavenumber is not a curiosity. It says how wide the convection cells will be.

A wavenumber a corresponds to a horizontal wavelength 2π/a, and a wavelength contains two counter- rotating rolls, so a single roll is π/a wide. In units of the layer depth, with a_c = π/√2, that is

$$\frac{\pi}{a_c} = \sqrt2 = 1.41421\ldots$$

Convection cells at onset are about 1.41 times as wide as the layer is deep. This is a prediction about a picture, it requires no measurement of any material property, and it is right: shallow layers of oil dusted with aluminium powder, the classic demonstration, produce cells of roughly that proportion.

The physical reading is a compromise. A very wide cell has to push fluid a long way horizontally against viscosity to lift a given amount; a very narrow one has a large horizontal temperature gradient and loses its buoyancy to conduction. The optimum sits between, and it sits near an aspect ratio of one because that is where neither penalty dominates — which is the sort of answer dimensional reasoning would guess and this calculation supplies exactly.

The case with no closed form, marked as borrowed #

The closed form exists only for stress-free boundaries — a layer with no shear at top and bottom, which is what an idealised layer between two free surfaces would have and what no experiment has.

A real experiment has rigid plates, where the velocity vanishes as well as the normal component. The eigenvalue problem is then a sixth-order boundary-value problem with no analytic solution, and its answer is

$$Ra_c = 1707.762, \qquad a_c = 3.117$$

The figure draws that value as a horizontal line in the colour this site reserves for a claim it did not derive, and says so in the label. It is two and a half times the free-free value, which is a large difference and an unsurprising one: rigid plates resist the horizontal motion the cells need, so more driving is required.

The mixed case — rigid below, free above, which is a pan of water — gives 1100.65. All three are eigenvalues of the same operator with different boundary conditions, and only the first has a closed form, which is a good illustration of how little the availability of a closed form has to do with the importance of a case.

Why linear theory gets the whole answer here #

The unusual thing about this problem is not that a threshold can be computed. It is that the computed threshold is the observed one, to within the accuracy of the experiment, which is not true of any other transition in this field.

The reason is the character of the bifurcation. Convection’s onset is supercritical: at the threshold the growing mode saturates at a small amplitude that rises continuously from zero as the Rayleigh number is raised, so the first thing that exists above threshold is a small steady flow with exactly the wavenumber linear theory predicted.

Compare pipe flow, where the bifurcation is subcritical and the observed transition depends on the size of the disturbance available. There is no finite-amplitude route to convection below Ra_c in a Boussinesq fluid — the state is genuinely stable, not merely linearly stable — so there is no receptivity problem and no dependence on how quiet the laboratory is.

That distinction is the most transferable thing in this essay. Whether a linear threshold is the observed threshold is decided by whether the bifurcation is supercritical, and it can be settled by a weakly nonlinear calculation rather than argued about.

The parameter that is missing from the answer #

There are two dimensionless groups in the problem and only one of them is in the threshold. The Rayleigh number is above; the other is the Prandtl number $\nu/\kappa$, which says which of the two diffusions is the faster, and it does not appear in $Ra_c$ or in $a_c$ anywhere.

That is exact rather than approximate, and the reason is worth having. The Prandtl number enters the linearised equations only multiplying the time derivatives, and at neutral stability the growth rate is zero, so every term it multiplies drops out. What is left is a purely spatial eigenvalue problem, and its answer belongs to the geometry and the boundary conditions alone.

The consequence is a strong and testable claim. Mercury, air, water and a heavy silicone oil have Prandtl numbers spanning five orders of magnitude — about 0.025, 0.7, 7 and several thousand — and all four begin to convect at the same Rayleigh number, in cells of the same proportions. The fluid decides what $Ra$ is for a given temperature difference; it does not get a say in what $Ra$ has to reach.

This holds because the instability sets in as a steady mode. If the marginal disturbance oscillated rather than standing still, the growth rate would be imaginary rather than zero, the time derivatives would survive, and the Prandtl number would be back in the threshold. That it does not is a theorem — the principle of the exchange of stabilities, established for this problem by Pellew and Southwell in the same 1940 paper that settled the boundary conditions — and it is a property of this problem rather than of stability theory in general.

Change the problem slightly and it fails. Rotate the layer, or run it in a magnetic field, and the first instability can become oscillatory — onset as a growing wave instead of a steady roll — and in the rotating case which of the two happens depends on the Prandtl number, so low-Prandtl liquid metals behave differently from water at the same rotation rate. The single-number answer is a privilege of the plainest version of the problem.

The sharpest failure is a layer with two diffusing components. Warm salty water over cool fresh water can be statically stable — denser at the bottom, so no parcel is buoyant — and convect anyway, because heat diffuses about a hundred times faster than salt: a descending parcel loses its warmth long before it loses its salt, and arrives heavier than its surroundings. The result is the centimetre-scale salt fingers and the thermohaline staircases found throughout the tropical ocean.

Which is a genuine limit on this essay’s opening picture. Buoyancy against two diffusions gives a threshold; buoyancy against two diffusions of two different things gives an instability where the density profile says there should be none.

What happens above the threshold, and how far this goes #

Raising the Rayleigh number takes the layer through a sequence, and the sequence is the reason this anchor continues.

Just above Ra_c: steady two-dimensional rolls, at the predicted wavelength.

Somewhat above: the rolls develop three-dimensional instabilities — cross-rolls, zigzag, skewed varicose — each with its own threshold, and the pattern becomes more complicated while remaining steady or periodic.

Higher still: time dependence, then aperiodic behaviour, then a state usually called turbulent convection with a thin boundary layer at each plate and a well-mixed interior.

At very high Rayleigh number — 10⁸ and beyond — the question becomes how the heat transport scales, and the answer is a subject of its own with two competing predictions and forty years of measurements.

Nothing above Ra_c is computed on this site. What the next rung does instead is take the truncation of this problem that Lorenz studied, which reduces the whole layer to three ordinary differential equations, and is honest about the fact that above r ≈ 5 it has stopped describing convection.

What “supercritical” costs to establish #

The claim that the bifurcation is supercritical is not a linear result, and it is worth saying where it comes from, because it is the load-bearing part of the argument that this threshold is the observed one.

The linear problem gives the threshold and the wavenumber and says nothing about amplitude: a growing mode grows exponentially for ever, which is the linearisation announcing its own expiry. To find out what the amplitude settles at, the expansion has to be carried to third order, and the coefficient of the cubic term decides the character.

A negative cubic coefficient saturates the growth, and the amplitude above threshold rises as the square root of the distance past it — continuous, with no jump and no hysteresis. That is supercritical, and it is what a Boussinesq layer has.

A positive coefficient means the cubic term drives the growth further, the amplitude jumps to whatever a higher-order term catches it at, and there is a range of parameters where two states are both stable. That is subcritical, and it is what produces hysteresis, sensitivity to disturbance size, and thresholds that depend on the laboratory.

The weakly nonlinear calculation is a page of algebra rather than a research problem, and it is available for most of the flows in this field. Where it has been done — Bénard convection, Taylor–Couette flow between rotating cylinders — the linear threshold turns out to be the observed one; where the answer comes out subcritical, as for plane Poiseuille flow, it does not. The character of the bifurcation, not the existence of the eigenvalue, is what decides whether a stability calculation predicts anything.

Where the model stops #

The Boussinesq approximation. The density is treated as constant everywhere except in the buoyancy term. That is excellent for a thin layer with a small temperature difference and it fails for a deep atmosphere, where compressibility matters and the relevant criterion involves the adiabatic lapse rate rather than a Rayleigh number.

Infinite horizontal extent. The analysis assumes the layer is unbounded sideways, so any wavenumber is available. A real container quantises the admissible wavenumbers and its threshold is therefore slightly above the ideal one, by an amount depending on the aspect ratio.

Linear. Everything here is a statement about infinitesimal disturbances. It happens to be the right statement, for the reason given above, and it is still a statement about a linearised problem.

Who found it, and when #

There is one more thing worth saying about why this problem, of all of them, has the clean answer. A shear flow’s instability draws its energy from the mean flow through the Reynolds stress working against the mean shear, which is a mechanism with the closure difficulty built into it from the start. Convection draws its energy from a potential — the buoyancy field — and a potential is a much simpler thing to write an eigenvalue problem about.

That is also why the analogous shear results are statements about which wavenumber grows fastest rather than about a threshold value: there is no potential to exceed, only a competition between rates.

Bénard published the experiments in 1900 and 1901 — thin layers of spermaceti and paraffin on a heated plate, with the hexagonal cells that carry his name.

Rayleigh gave the theory in 1916 and obtained 27π⁴/4 for the free-free case. The irony is that Bénard’s own experiments were dominated by surface tension rather than buoyancy, since his layers had a free upper surface and were less than a millimetre deep — so the theory Rayleigh built to explain Bénard’s cells explains a different phenomenon, and Pearson identified the surface-tension mechanism in 1958. The names have stayed attached in the way names do.

Jeffreys computed the rigid–rigid eigenvalue in 1928, and Pellew and Southwell gave the definitive treatment of all three boundary conditions in 1940.

Where the ladder goes next #

The next rung is the truncation. Three numbers left of a fluid takes this problem, keeps two Fourier modes of the streamfunction and one of the temperature, and studies what is left — which turns out to be one of the most consequential systems of equations in twentieth-century science and to have stopped describing convection long before it got interesting.