Brunt–Väisälä frequency
angular frequency at which a vertically displaced parcel will oscillate within a statically stable environment
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.
Derivation for a general fluid
Consider a parcel of water or gas that has density
ρ
0
{\displaystyle \rho _{0}}
. This parcel is in an environment of other water or gas particles where the density of the environment is a function of height:
ρ
=
ρ
(
z
)
{\displaystyle \rho =\rho (z)}
. If the parcel is displaced by an infinitesimally small vertical increment
z
′
{\displaystyle z'}
, and it maintains its original density so that its volume does not change, it will be subject to an extra gravitational force against its surroundings of:
ρ
0
∂
2
z
′
∂
t
2
=
−
g
[
ρ
(
z
)
−
ρ
(
z
+
z
′
)
]
{\displaystyle \rho _{0}{\frac {\partial ^{2}z'}{\partial t^{2}}}=-g\left[\rho (z)-\rho (z+z')\right]}
where
g
{\displaystyle g}
is the gravitational acceleration, and is defined to be positive. We distribute the negative preceding the substitution of the linear approximation:
ρ
(
z
+
z
′
)
−
ρ
(
z
)
=
∂
ρ
(
z
)
∂
z
z
′
{\displaystyle \rho (z+z')-\rho (z)={\frac {\partial \rho (z)}{\partial z}}z'}
and then dividing
ρ
0
{\displaystyle \rho _{0}}
to the RHS, giving:
∂
2
z
′
∂
t
2
=
g
ρ
0
∂
ρ
(
z
)
∂
z
z
′
{\displaystyle {\frac {\partial ^{2}z'}{\partial t^{2}}}={\frac {g}{\rho _{0}}}{\frac {\partial \rho (z)}{\partial z}}z'}
The above second-order differential equation has the following solution:
z
′
=
z
0
′
e
i
N
t
{\displaystyle z'=z'_{0}e^{iNt}}
where the Brunt–Väisälä frequency
N
{\displaystyle N}
is:
N
=
−
g
ρ
0
∂
ρ
(
z
)
∂
z
{\displaystyle N={\sqrt {-{\frac {g}{\rho _{0}}}{\frac {\partial \rho (z)}{\partial z}}}}}
For negative
∂
ρ
(
z
)
∂
z
{\displaystyle {\frac {\partial \rho (z)}{\partial z}}}
, the displacement
z
′
{\displaystyle z'}
has oscillating solutions (and N gives our angular frequency). If it is positive, then there is run away growth – i.e. the fluid is statically unstable.
“Brunt–Väisälä frequency” enters the record as angular frequency at which a vertically displaced parcel will oscillate within a statically stable environment. Crown Archives preserves that source wording while asking what Brunt, Väisälä and frequency can confirm, complicate or overturn.
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