Richardson Number (Buoyancy against Shear)

Richardson's number is the contest between stratification and stirring. Buoyancy wants a density-stratified fluid to stay in layers, heavy below light. Shear wants to overturn it. The ratio of the two — potential energy needed to mix, over kinetic energy available in the shear — is $Ri$, and it decides whether an estuary keeps its salt wedge, whether a plume punches through an inversion, whether a chilled supply layer stays on the floor of a room.

Small $Ri$: shear wins, Kelvin–Helmholtz billows roll up along the interface, and the layers mix. Large $Ri$: buoyancy wins, mixing across the interface becomes slow and intermittent, and the stratification persists. In between, both matter and neither dominates. The same statement appears in coastal and hydraulic work as the densimetric Froude number, $Fr_d = 1/\sqrt{Ri}$ — identical physics, inverted and square-rooted, because one trade thinks in terms of a wave speed and the other in terms of an energy ratio.

The 0.25 is the most oversold number in stratified flow, and it is worth being precise about what it does and does not say. Miles and Howard proved in 1961 that if the gradient Richardson number exceeds 1/4 everywhere in a steady, inviscid, parallel shear flow, that flow is stable to small disturbances. Every word in that sentence is load-bearing. It is a sufficient condition, not a necessary one. It concerns the gradient Richardson number, computed from local derivatives, not the bulk number this page computes from layer-averaged differences. And it says nothing about mixing rates in flow that is already turbulent.

A bulk Richardson number averages over a layer and can sit comfortably above 0.25 while a thin sublayer inside it is unstable and mixing hard. Field and laboratory measurements routinely show mixing at bulk values of 1 and more. Treat 0.25 as a signpost, never as a criterion you can defend in a report without saying which Richardson number you meant.

The length and the velocity are conventions, and here they are unusually slippery. $L$ may be the layer depth, the interface thickness, the jet width, or simply the distance between two instruments that happened to be available. $v$ is a velocity difference across that same distance, not a speed past a fixed point. Change what you call the layer and $Ri$ changes with it — linearly in $L$, as the square in $v$. Two engineers can compute Richardson numbers a decade apart for the same estuary and both be right about their own definitions. Always quote the pair, and never compare a Richardson number to a published threshold without checking that the threshold used the same ones.

For a gas at constant pressure there is a useful shortcut: $\Delta\rho/\rho \approx \Delta T/T$ in kelvin. A 6 K difference across a 293 K room is about 2 %, and that 2 % is the entire buoyant driving force displacement ventilation has to work with. It is also why a smoke layer in a fire is so much more robust — a few hundred kelvin gives a density difference of tens of percent, and the layer holds against far more stirring than a comfort-cooling stratification ever could.