Stratified Buoyancy Frequency

Worked Example

Consider an incompressible Boussinesq fluid model with a constant vertical density gradient, where the z-axis points upwards. The ambient fluid density is given by ρ(z) = ρ₀ + a·z. Let the reference density ρ₀ = 1000 kg/m³, and gravitational acceleration g = 9.81 m/s².

Suppose the density gradient is a = -2 kg/m⁴. The buoyancy frequency squared (N²) is calculated as N² = -(g/ρ₀)·a. Because N² > 0, the stratification is stable. A displaced fluid parcel will undergo simple harmonic motion with frequency N ≈ 0.14007 rad/s.

If we displace the parcel by A = 0.5 m at time t = 0 with zero initial velocity, its position at t = 5 s is z(t) = A·cos(N·t) = 0.5 * cos(0.14007 * 5) ≈ 0.382 m. Its acceleration is -N²·z(t) ≈ -0.0075 m/s², showing a restoring force acting downwards towards equilibrium.

1. Identify inputs: a = -2, displacement A = 0.5, time t = 5.
2. Calculate N² = -(g/ρ₀)·a = -(9.81 / 1000) * (-2) = 0.01962.
3. Find N = sqrt(0.01962) = 0.14007.
4. Calculate z(t) = A·cos(N·t) = 0.5 * cos(0.14007 * 5) = 0.3823.

Experiment

You can interact with these dynamics directly using the Stratified Buoyancy Frequency module.

1. Set the "Density gradient" control to -2. This creates a stable stratification where density decreases with height (a negative gradient when z points upwards).
2. Set the "Displacement" to 0.5 m to give the parcel an initial offset from its equilibrium depth.
3. Change the "Time" control to 5 s. Observe the calculated "Parcel displacement" which should match approximately 0.382 m.
4. Now change the "Density gradient" to 2 kg/m⁴. The value of N² becomes negative (-0.01962), indicating an unstable (overturning) stratification. The stability changes to "Unstable".
5. At t = 5 s in this unstable regime, note that the displacement grows rather than oscillates (0.627 m), and the acceleration is positive, driving the parcel further away from z=0.

Limitations

This model employs a local constant-gradient and small-displacement idealization. It assumes that the displaced fluid parcel does not mix with its surroundings or exchange heat or salt (adiabatic and isohaline displacement), and that its internal density remains fixed at its reference origin value ρ₀.

Furthermore, the constant density gradient ρ(z) = ρ₀ + a·z is a local approximation. This is not a claim of globally linear density over large depths, nor does it account for the compressibility of real fluids such as seawater or the atmospheric lapse-rate where adiabatic expansion significantly alters the parcel's density as it moves vertically. Real oceans and atmospheres often have non-linear stratification profiles. This concept is distinct from flotation or metacentric hull stability, which deals with solid objects displacing fluid at a free surface.

References

For a comprehensive treatment of seawater thermodynamics, stratification, and the definition of the Brunt-Väisälä (buoyancy) frequency in real oceanographic contexts, consult the official TEOS-10 documentation:

• TEOS-10 Manual: The international thermodynamic equation of seawater (2010). This manual describes the modern standard for calculating seawater properties, including the exact definitions for stability and buoyancy frequency beyond the Boussinesq approximation.