 Richardson number

A Richardson number compares stratification effects with an inertial or shear scale. For a stably stratified parallel shear flow $U(z)$, the gradient Richardson number iswhere $N$ is the buoyancy frequency. Stronger stable stratification resists vertical displacement, while shear can supply perturbation kinetic energy.

A bulk Richardson number replaces local derivatives by a buoyancy difference, layer thickness and characteristic velocity difference; the chosen normalization must be stated. Thus a bulk estimate and the minimum local ratio need not give the same diagnosis. The familiar quarter threshold belongs to a specific inviscid parallel-flow stability theorem: a sufficiently large gradient ratio everywhere can rule out growing normal modes under its assumptions. A local value below one quarter alone does not establish instability, and viscosity, boundaries or finite-amplitude disturbances require separate consideration.

The Richardson number (\(Ri\)) is a dimensionless number used in fluid mechanics and meteorology to quantify the relative importance of buoyancy compared to mechanical stirring (or shear) in a flow. It is especially relevant in the study of stratified fluids, such as in atmospheric and oceanic flows.