Richardson number

Builds on Dimensionless numbers in fluid mechanics Brunt–Väisälä frequency Boussinesq approximation for buoyancy

Definition

Consider a horizontal shear flow $U(z)$ in a stratified Boussinesq fluid with Brunt–Väisälä frequency $N(z)$ , where $z$ is the height. At a height where $U'(z) \ne 0$ , the gradient Richardson number is $\mathrm{Ri}(z) = \frac{N^2(z)}{\big( U'(z) \big)^2}.$ For a flow with a velocity difference $\Delta U \ne 0$ and a reduced gravity $g' = g \, \Delta\rho / \rho_0$ across a layer of thickness $h$ , the bulk Richardson number is $\mathrm{Ri}_b = g' h / (\Delta U)^2$ .

Remarks

Both forms are dimensionless, since $N$ and $U'$ are rates with dimension $\mathsf{T}^{-1}$ . The Miles–Howard theorem states that an inviscid stratified parallel shear flow with $\mathrm{Ri} \ge 1/4$ everywhere is linearly stable.

Example: $U(z) = S z$ with constant shear $S = 10^{-2}$ s $^{-1}$ and $N = 10^{-2}$ s $^{-1}$ gives $\mathrm{Ri} = 1$ , so the Miles–Howard criterion applies and the flow is linearly stable.

Non-example: the ratio $N / \abs{U'}$ is not the Richardson number; it is its square root, and the threshold $1/4$ would become $1/2$ .

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Boussinesq approximation for buoyancy concept
Brunt–Väisälä frequency concept
Dimensionless numbers in fluid mechanics concept
Richardson number definition

Tag	`YBSTEYPVDF`
URI	`m2://YBSTEYPVDF`
Kind	definition
Writing	W0 stub
MSC	76E20, 76F45, 76M55
Defines	Richardson number
Uses	Dimensionless numbers in fluid mechanics, Brunt–Väisälä frequency, Boussinesq approximation for buoyancy
Changed	3 November 2026

Richardson number

Definition

Remarks

Reading path

Further reading

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