Five numbers, one name

The number that stops the mixing computes the threshold by bisection on a parcel-exchange energy balance and gets exactly a quarter: swapping two parcels across a shear releases $(\Delta u)^2/4$ per unit mass and costs $N^2\delta^2$, and the two are equal when $N^2/(du/dz)^2 = 1/4$. Miles and Howard proved in 1961 that the same quarter is a rigorous sufficient condition for stability, and the coincidence between the crude argument and the rigorous bound is one of the tidier results in the subject.

None of that is in dispute. What this essay is about is the two words in front of it — “its” and “Richardson number” — because on any real flow there are five of them.

The flow, and the profile the number is a profile of #

Take the profile the entire stability literature is written about, and the one whose kink decides its own instability: a hyperbolic-tangent shear layer with a matching density step,

$$U(z) = U_0\tanh(z/h), \qquad \rho(z) = \rho_0 - \tfrac{\Delta\rho}{2}\tanh(z/h).$$

Both the buoyancy frequency and the shear then carry the same $\mathrm{sech}^2(z/h)$, and the gradient Richardson number is their ratio:

$$\mathrm{Ri}_g(z) = \frac{N^2}{S^2} = J\cosh^2(z/h), \qquad J = \frac{g\Delta\rho\,h}{2\rho_0U_0^2}.$$

That is not a number. It is a function of height, least at the centre of the layer and growing without bound away from it — because the shear disappears at the edges and the stratification does not.

Five ways to reduce it to a number #

Every one of these appears in print as “the Richardson number”, and every one is a defensible thing to compute.

The minimum gradient value, $J$. A local ratio at one height. This is what Miles’ and Howard’s theorem is about, since the theorem’s hypothesis is that $\mathrm{Ri}_g$ exceeds a quarter everywhere, which for this profile means at its minimum.

The bulk number on the layer, $g\Delta\rho(2h)/\rho_0(2U_0)^2$, formed from the layer’s own thickness and its own velocity difference. On this profile it equals $J$ exactly — a coincidence of this particular pair of profiles, and not a general identity.

The bulk number on the half-layer, $g\Delta\rho h/\rho_0U_0^2$, which is the same quantities with the halves not taken. Exactly twice the first.

The bulk number on the whole measured flow, formed from the density difference and the velocity difference across whatever was measured. For a domain of $\pm3h$ it is 3.02 times $J$, and it grows in proportion to the depth measured over.

The depth-averaged gradient number, the mean of $\mathrm{Ri}_g$ over the domain. For $\pm3h$ it is 17.3 times $J$, and since the integrand is a $\cosh^2$ it grows without limit as the domain widens — doubling the domain multiplies it by twenty-two.

The two that are honest and the three that are not #

Sorting them makes the situation clearer than listing them.

Local, and therefore meaningful. The minimum gradient number is a property of the flow at one height and does not change when the measurement is extended. Widening the domain from $\pm3h$ to $\pm6h$ leaves it unaltered to nine decimal places, which the assertion behind these figures checks.

Bulk, and therefore convention-dependent. The three bulk numbers differ by factors of two and three, and the factors come entirely from which thickness and which velocity difference were chosen. They are stable — a bulk number computed twice from the same definition gives the same answer — but they are not comparable across papers.

Averaged, and therefore meaningless. The depth average of a $\cosh^2$ has no limit. A paper that quotes it has quoted a property of its own domain, and two identical experiments in tanks of different depths would report Richardson numbers differing by an order of magnitude.

That last one is the sharpest case, and it is not a straw man: depth-averaged gradient Richardson numbers are computed routinely from atmospheric soundings and from ocean profiles, where the “domain” is whatever range the instrument covered.

The theorem runs one way, and it is quoted both ways #

The second half of the problem is the direction of the implication, and it is reversed so often that the reversal has become the standard statement.

What Miles and Howard proved: if $\mathrm{Ri}_g > 1/4$ everywhere, the flow is stable to infinitesimal disturbances.

What follows: instability requires $\mathrm{Ri}_g < 1/4$ somewhere.

What does not follow: that $\mathrm{Ri}_g < 1/4$ somewhere produces instability.

The distinction is not pedantic. Take the same tanh shear layer and displace the density interface away from the centre of the shear. The minimum gradient Richardson number can be made as small as desired while the flow remains stable, because the region where the number is small is no longer the region where a disturbance would grow. The theorem is silent about such a flow and the folklore version predicts instability.

A sufficient condition for stability is not a criterion for transition, and treating it as one is the same error as treating a term ratio’s value of one as a threshold — a statement about one side being read as a statement about both.

Why the profile matters more than the number #

The general statement behind all of this is that a stratified shear flow is characterised by two functions, not by one number, and every attempt to compress the two into one loses information that the stability problem needs.

The gradient Richardson number is already a compression: it is the ratio of two profiles, so two flows with entirely different $N^2(z)$ and $S^2(z)$ can share a $\mathrm{Ri}_g(z)$ and behave differently. Compressing further, to a single number, loses the shape of even that ratio.

That is why the stability literature works with the Taylor–Goldstein equation — an eigenvalue problem that takes both profiles as input and returns a growth rate — rather than with a criterion. The quarter survives as a bound because a bound is all it ever was.

What survives, and it is quite a lot #

It would be a bad essay that left the impression that the Richardson number is useless. Three things survive intact and they are the three the number is worth having for.

The scaling. Every threshold in this subject is a Richardson number, and a Richardson number is a statement about $g\Delta\rho h/\rho_0 U^2$ as a whole. Change the fluid, the depth or the shear and the numbers do not move; the conditions move, exactly as the group requires. That is the whole content of a dimensionless group and it is unaffected by any of the above.

The sign. A negative Richardson number is an unstably stratified flow, which convects, and the sign is convention-independent. Every one of the five agrees about it.

And the ordering. Two flows compared under the same convention are ranked correctly, and it is comparison across conventions that fails. A single research group using a single definition throughout produces internally consistent results and always has.

What does not survive is the number as an absolute. The quarter is a bound on one specific quantity, and quoting a threshold from one convention against a measurement from another is an error of a factor between two and seventeen.

What this does to a measurement #

The practical consequence is about comparing numbers across sources, and it is severe.

A field campaign reporting “Richardson numbers of 0.2 in the observed layers” has said something that could mean any of the five quantities above. If it is the minimum gradient value the layers are below the bound and could be unstable; if it is a bulk number on the whole sounding it corresponds to a minimum gradient value of about 0.07, well below; and if it is a depth average it corresponds to about 0.012, which would be a violently unstable layer.

A factor of seventeen in the reported number is a factor of seventeen in the conclusion, and the convention is very often not stated.

The remedy is the one this collection applies to every group it draws: print which quantity was formed, and from which lengths. A figure that says “Ri = 0.2” is not a figure; a figure that says “minimum gradient Richardson number 0.2, on a layer of half-thickness $h$” is.

The convention that is nearly always meant #

Worth ending the diagnostic half with a recommendation, because the situation is fixable and the fix costs one sentence.

For a stability question, the quantity to compute is the minimum of the gradient Richardson number over the profile, because that is the quantity the theorem is about and the only one that does not move when the measurement domain does. For a mixing question, the quantity is the flux Richardson number, which is not obtainable from mean profiles at all. For a scaling or a regime map, any consistent bulk definition will do, provided it is stated.

The failure mode is not that people choose badly. It is that the choice is usually invisible, because “the Richardson number” reads like the name of a quantity rather than the name of a family — in exactly the way “the Reynolds number” does, and for exactly the same reason.

The sixth one, which is the one that decides the mixing #

The list of limitations names a sixth quantity of the same name and sets it aside, and it deserves better, because it is the member of the family that carries the practical weight and it repeats both of this essay’s failures at once.

The flux Richardson number is not a ratio of gradients at all. It is a ratio of two terms in the turbulent kinetic-energy budget: the rate at which buoyancy removes energy from the turbulence, divided by the rate at which the shear supplies it. So it measures what a turbulent stratified flow is doing rather than what its mean profiles look like, and it cannot be obtained from a sounding — it needs the fluxes themselves.

Its importance is that it converts a measurement of turbulence into a mixing rate. Rearranged as the ratio of buoyancy flux to dissipation — conventionally the mixing efficiency — it is the factor that turns a measured dissipation rate into a diapycnal diffusivity, through

$$K_\rho = \Gamma\,\frac{\varepsilon}{N^2}.$$

That single relation is how essentially every published estimate of ocean interior mixing has been made. A profiler measures $\varepsilon$ from microstructure shear and $N^2$ from the density profile, and $\Gamma$ supplies the rest.

The value used is 0.2, and its provenance is exactly this essay’s second complaint. Osborn derived it in 1980 as an upper bound — the most of the shear production that could go into buoyancy rather than into dissipation — and it has been used as an equality ever since. A one-way result quoted both ways, in a different quantity, with the same consequence: a bound treated as a value.

And it repeats the first complaint too. Measurements and simulations since have found the efficiency varying by a factor of several — with the turbulence’s own intensity, with the Richardson number itself, and with whether the layer is actively overturning or decaying — so it is neither a constant nor a property of the fluid. A quantity that ranges from near zero to about a third is being used as 0.2 throughout an entire field’s inventory of ocean mixing.

Which puts the essay’s argument in its most consequential form. Of the six quantities called the Richardson number, one carries a theorem and is local, three are conventions, one is an artefact of a domain, and the sixth is a bound used as a constant in a calculation that sets the diffusivity of the world’s oceans. None of them can be substituted for another, and the only thing they reliably share is a name.

The same disease elsewhere #

This is not a defect peculiar to stratified flow. It is what happens to every dimensionless group whose defining lengths are not unique, and this collection has met it repeatedly.

The Reynolds number on a cylinder’s radius and on its diameter differ by two, and the transition values quoted in different books differ by exactly that.

The Knudsen number on a channel’s gap and on its half-gap differ by two, and every rung boundary of the rarefaction ladder is quoted to one significant figure.

The Womersley number is unambiguous, because a tube has one radius — which is why the profile it governs has the best-behaved threshold on the site — which is why it has a better reputation than either of the above and is quoted with more confidence than it has earned.

The rule that emerges is simple and is not usually stated: a group whose lengths are unique can be compared across sources, and one whose lengths are a choice cannot.

What the picture cannot show #

These are stated profiles, not solutions. A tanh shear layer with a matching tanh density is a model chosen because the stability literature uses it and because it has closed forms; nothing in this collection solves for it, and the numbers above are arithmetic applied to a stated shape.

The flux Richardson number is absent. A sixth quantity of the same name is the ratio of the buoyancy flux to the shear production in a turbulent flow, which is a property of the turbulence rather than of the mean profiles and needs a closure to compute. It is the one that appears in mixing parameterisations, and it is not obtainable from any of the five here.

The Taylor–Goldstein problem is not solved here. Whether this profile actually goes unstable below $J = 1/4$ — it does, and the boundary is exactly a quarter for this particular pair of profiles, which is Holmboe’s result — is an eigenvalue calculation this collection has not built.

And nothing here is turbulent. The quarter is about the onset of a linear instability. What happens afterwards, how much mixing results and what the eventual state is, are questions the criterion cannot answer and is routinely asked.

Who found it, and when #

Richardson introduced his number in 1920, in a paper about atmospheric turbulence and eddy diffusion, and the quantity he wrote was the local gradient ratio. Taylor and Goldstein set up the stability problem in 1931. Miles and Howard proved the sufficiency of the quarter in 1961, in adjacent papers in the same journal, and Howard’s proof is famously three pages long and entirely elementary.

The proliferation of conventions came afterwards and came from measurement. A sounding gives temperature and wind at discrete levels, and a gradient computed between two levels is a bulk number over the interval between them — so the “gradient” Richardson number computed from real data is always a bulk one with an interval attached, and the interval is the instrument’s rather than the flow’s.

The surprising connection is that this makes the reported number depend on the resolution of the measurement. A finer sounding resolves sharper gradients, and a sharper shear raises $S^2$ faster than $N^2$, so the same atmosphere reports a smaller Richardson number when measured more finely. That is not a small effect, and it is the same hazard as a grid that shares its answer’s symmetry: refining a sounding from 100 m to 10 m can halve the reported values, and the layers that appear unstable at fine resolution are invisible at coarse. A criterion whose value depends on how carefully somebody looked is exactly the failure mode this collection’s account of thresholds is about — arriving here not through a tolerance, but through a measurement.

Where the ladder goes next #

Above this rung is the Taylor–Goldstein eigenvalue problem, which takes both profiles and returns a growth rate rather than a verdict — the honest form of the question, and one needing an eigenvalue solver of the kind the convection onset uses.

Beside it sits the quarter itself, computed from an energy balance, and the inflection criterion, which has the same one-way logic and the same reputation for being a criterion. Below it is the general account of what a group’s own value is worth, in which this is the row that fails for a different reason from all the others.