where is gravity, is density, is a representative flow speed, and 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
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.
In aviation, the Richardson number is used as a rough measure of expected air turbulence. A lower value indicates a higher degree of turbulence. Values in the range 10 to 0.1 are typical, with values below unity indicating significant turbulence.
where g is the gravitational acceleration, is the thermal expansion coefficient, Thot is the hot wall temperature, Tref is the reference temperature, L is the characteristic length, and V is the characteristic velocity.
Typically, the natural convection is negligible when Ri < 0.1, forced convection is negligible when Ri > 10, and neither is negligible when 0.1 < Ri < 10. It may be noted that usually the forced convection is large relative to natural convection except in the case of extremely low forced flow velocities. However, buoyancy often plays a significant role in defining the laminar-turbulent transition of a mixed convection flow. In the design of water filled thermal energy storage tanks, the Richardson number can be useful.
In oceanography, the Richardson number has a more general form which takes stratification into account. It is a measure of relative importance of mechanical and density effects in the water column, as described by the Taylor-Goldstein equation, used to model Kelvin-Helmholtz instability which is driven by sheared flows.
where N is the Brunt-Väisälä frequency.
The Richardson number defined above is always considered positive. A negative value of N² (i.e. complex N) indicates unstable density gradients with active convective overturning. Under such circumstances the magnitude of negative Ri is not generally of interest. It can be shown that Ri < 1/4 is a necessary condition for velocity shear to overcome the tendency of a stratified fluid to remain stratified, and some mixing (turbulence) will generally occur. When Ri is large, turbulent mixing across the stratification is generally suppressed.