Richardson number

The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow shear term:

${\displaystyle \mathrm {Ri} ={\frac {\text{buoyancy term}}{\text{flow shear term}}}={\frac {g}{\rho }}{\frac {\partial \rho /\partial z}{(\partial u/\partial z)^{2}}}}$

where ${\displaystyle g}$ is gravity, ${\displaystyle \rho }$ is density, ${\displaystyle u}$ is a representative flow speed, and ${\displaystyle z}$ is depth.

The Richardson number, or one of several variants, is of practical importance in weather forecasting and in investigating density and turbidity currents in oceans, lakes, and reservoirs.

When considering flows in which density differences are small (the Boussinesq approximation), it is common to use the reduced gravity g' and the relevant parameter is the densimetric Richardson number

${\displaystyle \mathrm {Ri} ={\frac {-\partial g'/\partial z}{(\partial u/\partial z)^{2}}}}$

which is used frequently when considering atmospheric or oceanic flows.

If the Richardson number is much less than unity, buoyancy is unimportant in the flow. If it is much greater than unity, buoyancy is dominant (in the sense that there is insufficient kinetic energy to homogenize the fluids).

If the Richardson number is of order unity, then the flow is likely to be buoyancy-driven: the energy of the flow derives from the potential energy in the system originally.