Thayer Watkins
Silicon Valley
& Tornado Alley

The Brunt-Väisälä Buoyancy Frequency
of the Atmosphere

Consider an infinitesimal parcel of air of area dA and height dz which is embedded in a column of air in hydrostatic balance. Let p(z) and ρ(z) denote the pressure and density in the parcel and p0(z) and ρ0(z) the pressure and temperature in the column.

The parcel pressure and density are assumed to be equal to to those of the column at one particular height z0. But if the parcel is moved adiabatically to a different level the pressure and density of the parcel may deviate from those of its environment in the column.

The parcel at some height z would be subject to two forces, gravity and the pressure forces from the environmental atmosphere. By Newton's Second Law

(ρdAdz)(d2z/dt2) = -(ρdAdz)g - dA(∂p0/dz)dz
which after division by the parcel volume dAdz reduces to
ρ(d2z/dt2 = -ρg - (∂p0/dz)

Because the column of air is in hydrostatic balance

(∂p/dz)0 = -ρ0g
and thus the force balance equation
for the parcel is
ρ( d2z/dt2) = -ρg + ρ0g
d2z/dt2 = -g(ρ-ρ0)/ρ

At the equilibrium height of the parcel z0, ρ and ρ0 are equal. Suppose that the parcel always adjusts to the local environmental pressure so that buoyancy is due only to the difference in density of the parcel compared to the environment. Since by the ideal gas law

(ρ-ρ0)/ρ = [p/RT - p/RT0]/(p/RT)
= [1 - T/T0]

But T/T0 = θ/θ0 since the pressures are equal. Thus

d2z/dt2 = g[1 - θ/θ0]

For the moment designate the RHS of the above equation as a(z). This acceleration a(z) is zero at z0 so by a Taylor's series expansion

a(z) = (∂a/∂z)z=z0(z-z0)

The potential temperature of the parcel θ is constant because the parcel adjusts adiabatically. Therefore the change in a(z) with height is due entirely to the variation in θ0 with height.

(∂a/∂z) = -(θ/θ02)(dθ0/dz)
which at z=z0 where θ=θ0 reduces to
(∂a/∂z) = -(1/θ0)(dθ0/dz) = -d(ln θ0)/dz

Thus the acceleration equation is

d2z/dt2 = g[d(ln θ0)/dz](z-z0)

This is the well-known differential equation for harmonic motion about an equilibrium at z0 and the frequency of oscillation is therefore given by N where

N2 = g[d(ln θ0)/dz]

N is called the Brunt-Väisälä buoyancy frequency.

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