Richardson Number Calculator

The Richardson number, Ri, is a dimensionless quantity that compares the stabilizing effect of buoyancy (density stratification) to the destabilizing effect of velocity shear in a fluid. A low Ri means shear dominates and the flow is prone to turbulence; a high Ri means buoyancy dominates and the flow stays stably stratified.

Ri = g(Δρ/ρ)L/V², where g is gravitational acceleration (9.81 m/s²), Δρ is the density difference between two layers, ρ is a reference density, L is a characteristic vertical length scale, and V is a characteristic velocity or shear scale.

Ri_c = 0.25 is the threshold from the Miles-Howard theorem: when the local Richardson number stays below 0.25 throughout a shear layer, the flow is unstable to small perturbations and can develop Kelvin-Helmholtz billows that mix the layers. Above 0.25, shear alone cannot overcome the stabilizing buoyancy, though the flow may still mix intermittently.

Kelvin-Helmholtz instability is the rolling, wave-like breakdown of a sheared interface between two fluid layers of different density, producing the characteristic billow or 'cat's eye' pattern seen in clouds and ocean interfaces. It is the mechanism that a low Richardson number predicts will occur.

Meteorologists compute the bulk Richardson number for atmospheric layers to forecast clear-air turbulence, low-level jet mixing, and whether a nocturnal temperature inversion will stay stably stratified or break down as wind shear increases overnight.

The gradient Richardson number uses local, continuous derivatives of density and velocity at a single point, while the bulk Richardson number (used by this calculator) uses finite differences between two layers, Δρ and a characteristic velocity scale V, making it easier to compute from typical field or model data.

Yes. A negative Ri occurs when the density difference term is unstably arranged (denser fluid sits above lighter fluid), meaning the layer is convectively unstable and will overturn from buoyancy alone, independent of any shear.

The Richardson number is closely related to the inverse square of a densimetric Froude number: Ri is proportional to 1/Fr². Where the Froude number is large (fast, shallow, gravity-dominated flow), the Richardson number is small, and vice versa.

Use the density of the lower, denser layer as the reference density in most oceanographic and limnological applications, or the ambient background density in atmospheric applications. Consistency matters more than the exact choice, since ρ only rescales the buoyancy term.

When buoyancy (from strong density stratification) is much larger than the kinetic energy available from shear, any turbulent eddy that tries to lift denser fluid over lighter fluid does work against gravity and loses energy faster than shear can supply it, so turbulence cannot be sustained and the flow relaminarizes.

Yes. Engineers use Ri to design stratified thermal storage tanks, selective withdrawal structures at reservoir outlets, and ventilation systems where controlling whether a density interface mixes or stays intact is the design goal.