Brunt–Väisälä Frequency from the Potential Temperature Gradient

A stable layer rings like a spring. Push a parcel of air upward in a layer where the potential temperature rises with height, and it finds itself cooler and denser than its surroundings, so it sinks back, overshoots and bobs up and down. The Brunt-Väisälä frequency $N$ is the angular frequency of that bobbing, in radians per second. The formula $N = \sqrt{\frac{g}{\theta}\,\frac{\Delta\theta}{\Delta z}}$ takes gravity $g$, the mean potential temperature $\theta$ of the layer, and the rise $\Delta\theta$ of potential temperature across a layer of depth $\Delta z$.

Reading the answer. The stability sits in $\Delta\theta/\Delta z$: the faster potential temperature climbs with height, the stiffer the spring and the larger $N$. Typical values are 0.01 per second in the troposphere, a period of about ten minutes, and twice that in the stratosphere. If the potential temperature falls with height, the layer is unstable and there is no oscillation, because the quantity under the root is negative. Gravity waves and mountain waves ring at this frequency, and the period of the bobbing is $2\pi/N$.

Worked cases. A rise of 3 K per kilometre at $\theta = 300$ K gives 0.0099 per second, which is about 0.01, the quoted tropospheric value. For an isothermal atmosphere at 250 K a different published expression, $N = g/\sqrt{c_p T}$, gives 0.0195644 per second. A layer built to match it, with $\theta = 300$ K and a rise of 11.70943 K over 1,000 m (a gradient of 0.0117094 K/m), returns the same value from the formula above, and run backward the formula recovers the 11.709 K rise or the 1,000 m depth.

Where it comes from. The form $N = \sqrt{(g/\theta)\,d\theta/dz}$ is given in Emanuel's Atmospheric Convection (1994), and Vallis's Atmospheric and Oceanic Fluid Dynamics (2017) is the other source. The page uses one gradient across the whole layer, so it describes the layer's average stability, not any sharper structure inside it. The potential-temperature page supplies $\theta$ at each level, and the Rossby-number page is the other dimensionless way meteorologists judge whether rotation or buoyancy matters for a flow.