applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

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
p_{0}(z)
and ρ_{0}(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 z_{0}. 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)(d^{2}z/dt^{2}) = -(ρdAdz)g - dA(∂p_{0}/dz)dz
which after division by the parcel volume dAdz reduces to
ρ(d^{2}z/dt^{2} = -ρg - (∂p_{0}/dz)

Because the column of air is in hydrostatic balance

(∂p/dz)_{0} = -ρ_{0}g
and thus the force balance equation for the parcel is
ρ( d^{2}z/dt^{2}) = -ρg + ρ_{0}g
hence
d^{2}z/dt^{2} = -g(ρ-ρ_{0})/ρ

At the equilibrium height of the parcel z_{0}, ρ and
ρ_{0} 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

But T/T_{0} = θ/θ_{0} since the pressures
are equal. Thus

d^{2}z/dt^{2} = g[1 - θ/θ_{0}]

For the moment designate the RHS of the above equation as a(z).
This acceleration a(z) is zero at z_{0} so by a Taylor's series
expansion

a(z) = (∂a/∂z)_{z=z0}(z-z_{0})

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 θ_{0} with height.

(∂a/∂z) = -(θ/θ_{0}^{2})(dθ_{0}/dz)
which at z=z_{0} where θ=θ_{0} reduces to
(∂a/∂z) = -(1/θ_{0})(dθ_{0}/dz)
= -d(ln θ_{0})/dz

Thus the acceleration equation is

d^{2}z/dt^{2} = g[d(ln θ_{0})/dz](z-z_{0})

This is the well-known differential equation for harmonic motion
about an equilibrium at z_{0}
and the frequency of oscillation is therefore given by N where