= Richardson number
{c}
{title2=$\mathrm{Ri}$}
{wiki}
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> is
$$
\operatorname{Ri}_g=\frac{N^2}{(dU/dz)^2},
$$
where $N$ 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.
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