Richardson number

The Richardson number, denoted Ri, is named after Lewis Fry Richardson (1881–1953). It is a dimensionless number that expresses the ratio of the buoyancy term to the flow shear term in fluid dynamics: $${\displaystyle \mathrm {Ri} ={\frac {\text{buoyancy}}{\text{flow shear}}}={\frac {g}{\rho }}{\frac {\partial \rho /\partial z}{(\partial u/\partial z)^{2}}},}$$ where ${\displaystyle g}$ is the local acceleration due to gravity, ${\displaystyle \rho }$ is the mass density, ${\displaystyle u}$ is a representative flow velocity, and ${\displaystyle z}$ is depth.

The Richardson number given above is one of several variants, and is of practical importance in weather forecasting as well as the investigation of density and turbidity currents in oceans, lakes, and reservoirs.

When considering flows in which density differences are small (the Boussinesq approximation), it is commonplace to use the reduced gravity ${\displaystyle g'}$. This situation gives the densimetric Richardson number $${\displaystyle \mathrm {Ri} =-{\frac {\partial g'/\partial z}{(\partial u/\partial z)^{2}}},}$$ which is frequently used when examining atmospheric or oceanic flows.

If Ri ≪ 1, buoyancy can be neglected in the flow. By contrast, if Ri ≫ 1, then buoyancy dominates in the sense that there is insufficient kinetic energy to homogenize the fluid. However, if Ri ≅ 1, then the flow is likely to be buoyancy-driven; that is, the energy of the flow derives from the potential energy of the system.