In atmospheric dynamics, oceanography, asteroseismology and geophysics, the Brunt–Väisälä frequency, or buoyancy frequency, is a measure of the stability of a fluid to vertical displacements such as those caused by convection. More precisely it is the frequency at which a vertically displaced parcel will oscillate within a statically stable environment. It is named after David Brunt and Vilho Väisälä. It can be used as a measure of atmospheric stratification.
Consider a parcel of water or gas that has density
\rho0
\rho=\rho(z)
z'
\rho0
\partial2z' | |
\partialt2 |
=-g\left[\rho(z)-\rho(z+z')\right]
where
g
\rho(z+z')-\rho(z)=
\partial\rho(z) | |
\partialz |
z'
\rho0
\partial2z' | |
\partialt2 |
=
g | |
\rho0 |
\partial\rho(z) | |
\partialz |
z'
The above second-order differential equation has the following solution:
z'=z'0ei
where the Brunt–Väisälä frequency
N
N=\sqrt{-
g | |
\rho0 |
\partial\rho(z) | |
\partialz |
For negative
\partial\rho(z) | |
\partialz |
z'
For a gas parcel, the density will only remain fixed as assumed in the previous derivation if the pressure,
P
N\equiv\sqrt{
g | |
\theta |
d\theta | |
dz |
\theta
g
z
Since
\theta=T
R/cP | |
(P | |
0/P) |
P0
N2\equivg\left\{
1 | |
T |
dT | - | |
dz |
R | |
cP |
1 | |
P |
dP | \right\}=g\left\{ | |
dz |
1 | |
T |
dT | - | |
dz |
\gamma-1 | |
\gamma |
1 | |
P |
dP | |
dz |
\right\}
\gamma=cP/cV
N2
N2\equivg\left\{
1 | |
\gamma |
1 | |
P |
dP | - | |
dz |
1 | |
\rho |
d\rho | \right\}=g\left\{ | |
dz |
1 | |
\gamma |
dlnP | - | |
dz |
dln\rho | |
dz |
\right\}
This version is in fact more general than the first, as it applies when the chemical composition of the gas varies with height, and also for imperfect gases with variable adiabatic index, in which case
\gamma\equiv\gamma01=(\partiallnP/\partialln\rho)S
S
If a gas parcel is pushed up and
N2>0
N2=0
N2<0
N2
N2
The Brunt–Väisälä frequency commonly appears in the thermodynamic equations for the atmosphere and in the structure of stars.
In the ocean where salinity is important, or in fresh water lakes near freezing, where density is not a linear function of temperature:where
\rho
The concept derives from Newton's Second Law when applied to a fluid parcel in the presence of a background stratification (in which the density changes in the vertical - i.e. the density can be said to have multiple vertical layers). The parcel, perturbed vertically from its starting position, experiences a vertical acceleration. If the acceleration is back towards the initial position, the stratification is said to be stable and the parcel oscillates vertically. In this case, and the angular frequency of oscillation is given . If the acceleration is away from the initial position, the stratification is unstable. In this case, overturning or convection generally ensues.
The Brunt–Väisälä frequency relates to internal gravity waves: it is the frequency when the waves propagate horizontally; and it provides a useful description of atmospheric and oceanic stability.