Richardson number ($\mathrm{Ri}$)

A Richardson number compares gravitational stratification with inertial or shear effects. Its precise form depends on the characteristic scales or local gradients being compared.

The ratio of stable buoyancy destruction to shear production of turbulent kinetic energy. With down-gradient transport and turbulent Prandtl number ${\mathrm{Pr}}_t$, it obeys ${\mathrm{Ri}}_f={\mathrm{Ri}}_g/{\mathrm{Pr}}_t$, rather than automatically equaling the gradient Richardson number. Unstable buoyancy production gives a negative value in this convention.

With positive stable buoyancy destruction $B$ and positive dissipation $ϵ$, this normalized sink fraction need not equal the Flux Richardson number $B/P$ away from equilibrium. If $R=(B+ϵ)/P$, then $J={\mathrm{Ri}}_f/R$. In the fixed-correlation equilibrium model for stratified turbulence, a minimum of $R=a/q+bq$ has $B=ϵ$, hence $J=1/2$. The actual flux Richardson number there is $R_{\min }/2$ and equals $1/2$ only when that minimum also satisfies the equilibrium condition $R=1$.

A local ratio of stabilizing buoyancy to shear across an interface of thickness $\delta$, using reduced gravity jump $g^{\prime }$ and characteristic velocity $u$. The dimensionless value is ${\mathrm{Ri}}_i=g^{\prime }\delta /u^2$. It is distinct from a whole-layer bulk Richardson number based on the total mixed depth. Local entrainment laws use this ratio when the transition thickness and velocity characterize the turbulence-producing region.

Combine a fixed-energy mixed-layer relation $u^{2}h=A{\Omega }^2{R}^3$, jump $g^{\prime }=N^{2}h/2$, and stress power proportional to $u^3$ with an energy-transfer factor $(\Omega /N{)}^{\alpha }(\delta /R{)}^{\beta }$. Eliminating depth in favor of the interfacial Richardson number gives $E\propto (\Omega /N{)}^{\alpha -1}(\delta /R{)}^{\beta +3/2}{\mathrm{Ri}}_i^{-3/2}$. A local-only entrainment closure requires $\alpha =1$, $\beta =-3/2$ and exponent $3/2$ on the Richardson number.

A bulk Richardson number compares a characteristic buoyancy difference across a layer with the squared characteristic flow speed. It uses large-scale differences rather than the pointwise gradient Richardson number. Its normalization depends on the chosen velocity, density and length scales.

For a stratified parallel shear flow, the gradient Richardson number compares the squared buoyancy frequency with squared vertical shear. The Miles–Howard theorem excludes exponentially growing inviscid normal modes when this ratio is at least $1/4$ everywhere. Where $U^{\prime }=0$, the equivalent criterion is written directly as $N^2\ge (U^{\prime }{)}^2/4$.