Brunt-Väisälä Frequency and Stability Condition

The Brunt-Väisälä frequency, denoted by N, is a fundamental concept in fluid dynamics and atmospheric science that describes the natural frequency at which a parcel of fluid displaced vertically will oscillate due to buoyancy if the surrounding fluid is statically stable.

Stability Analysis Based on N²

The stability of the atmosphere (or any stratified fluid) is determined by the sign of N²:

• Static Stability ($N^2 > 0$): If N² is positive, buoyancy acts as a restoring force, causing the displaced parcel to oscillate around its equilibrium position.
• Static Neutrality ($N^2 = 0$): If N² is zero, there is no restoring or destabilizing force. The parcel remains in its new position after displacement.
• Static Instability ($N^2 < 0$): If N² is negative, buoyancy acts as an amplifying force, pushing the displaced parcel further away from its equilibrium position.

Condition for Instability

The question states that $N^2 < 0$. In this scenario, the square of the Brunt-Väisälä frequency is negative. This indicates that there is no restoring force acting on a displaced air parcel. Instead, the buoyant force will cause the parcel to accelerate away from its original position, leading to a condition of static instability.

Therefore, any air parcel with $N^2 < 0$ will be statically unstable.