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BruntVäisälä frequency
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If a medium exhibits stratification, meaning it consists of layers with different densities, there is a possibility that the density difference becomes unstable, causing the layers to mix and the system to become homogeneous.
As long as the system remains stable, any disturbance will lead to oscillations that dissipate over time. The frequency associated with this behavior is known as the Brunt-Väisälä frequency, which exists in both the atmosphere and the ocean.
The following video shows a system with two different densities, where a cork is placed and oscillates in response to a disturbance, maintaining the order between the stable layers:
ID:(11754, 0)
Vaisala-Brunt Oscillation
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The energy associated with the Coriolis force can be estimated by considering the Coriolis force and a characteristic length $L$. The Coriolis force is the product of mass $m$, the Coriolis factor $f$, and velocity $U$. On the other hand, the energy associated with the inertial force is simply the kinetic energy proportional to $mU^2$.
Based on this, the Rossby number is defined as:
$R_0 = \displaystyle\frac{m U^2}{ m f U L}$
Thus, the Rossby number represents the ratio between the fluid's kinetic energy and the effect of the Coriolis force.
| $ R_0 =\displaystyle\frac{ U }{ f R }$ |
(ID 11753)
For the case in which the Rossby number
| $ R_0 =\displaystyle\frac{ U }{ f R }$ |
with $U$ representing velocity, $f$ as the Coriolis factor, and $L$ as a characteristic length, which is of the order of unity, we can determine that the characteristic length is approximately given by:
$L \sim \displaystyle\frac{U}{f}$
The velocity $U$ can be modeled using the Brunt-V is l frequency
| $ N = \sqrt{\displaystyle\frac{ g }{ \theta }\displaystyle\frac{ \Delta\theta }{ \Delta z }}$ |
where $g$ is the gravitational acceleration, $\Delta\theta/\theta$ represents the variation in potential temperature, and $\Delta z$ is the variation in height. In this case, the velocity can be expressed as:
$U\sim H N$
where $H$ denotes the height. Thus, the characteristic size can be obtained as:
| $ \lambda_R = \displaystyle\frac{ N H }{ f }$ |
(ID 11760)
Examples
If a medium exhibits stratification, meaning it consists of layers with different densities, there is a possibility that the density difference becomes unstable, causing the layers to mix and the system to become homogeneous.
As long as the system remains stable, any disturbance will lead to oscillations that dissipate over time. The frequency associated with this behavior is known as the Brunt-V is l frequency, which exists in both the atmosphere and the ocean.
The following video shows a system with two different densities, where a cork is placed and oscillates in response to a disturbance, maintaining the order between the stable layers:
(ID 11754)
ID:(1525, 0)