Richardson Number

The Richardson Number (Ri) is a dimensionless ratio that measures whether wind shear or atmospheric stability dominates at a given altitude. When Ri falls below 0.25, shear wins and the air becomes turbulent.

How It Works#

The atmosphere resists vertical mixing when it is statically stable. Warm, buoyant air wants to stay above cooler, denser air. Wind shear, the change in wind speed or direction with altitude, pushes back against that stability.

The Richardson Number compares those two forces directly. Its formula is:

$Ri = \frac{g / \theta \cdot d\theta/dz}{(du/dz)^2}$

Here, $g$ is gravitational acceleration, $\theta$ is potential temperature, $d\theta/dz$ is the rate of temperature change with altitude, and $du/dz$ is the rate of wind speed change with altitude. A large numerator means strong stability. A large denominator means strong shear.

When Ri exceeds 1.0, the atmosphere is strongly stable and suppresses mixing. Between 0.25 and 1.0, conditions are marginal. Below 0.25, shear overcomes stability and turbulence develops. This threshold of 0.25 is the critical Richardson Number, often written as $Ri_c$.

Example in Aviation#

A jet stream passes over a mountain range at FL350. Temperatures are cold and the atmosphere appears stable on a standard sounding. However, wind speed increases sharply over just a few hundred feet of altitude, creating strong shear.

A forecaster calculates Ri for that layer and finds a value of 0.15. This falls well below the $Ri_c$ threshold. The forecast team issues a SIGMET for severe clear-air turbulence (CAT), the kind of turbulence that appears without any visible cloud or storm. An airliner crossing the area reports moderate to severe chop, exactly as predicted.

Why It Matters#

Pilots rarely calculate Ri themselves. Dispatchers, meteorologists, and numerical weather models do that work. But understanding what the number represents helps pilots interpret turbulence forecasts and take SIGMETs seriously, especially when skies are clear and no convective weather is visible.

CAT is responsible for the majority of serious turbulence injuries in commercial aviation. It is invisible, sudden, and strongest near jet streams and mountain wave regions. The Richardson Number is one of the primary diagnostic tools forecasters use to locate and predict it.

Key Takeaways#

• The Richardson Number compares atmospheric stability against wind shear strength.
• Values below 0.25 indicate turbulent conditions; above 1.0 means strongly stable air.
• The critical threshold is $Ri_c = 0.25$, derived from fluid dynamics theory.
• Low Ri values near jet streams are a primary indicator of clear-air turbulence.
• Numerical weather models use Ri to generate turbulence forecasts and SIGMETs.