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BruntVäisälä frequency
Video 
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:
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Potential temperature
Equation
The vertical stability in the atmosphere depends on both temperature $T$ and pressure $p$ variations. Therefore, it is useful to work with a single parameter called potential temperature to model instability. Potential temperature is defined as:
 $ \theta = T \left(\displaystyle\frac{ p_0 }{ p }\right)^{ R / c_p }$ |
where $R$ is the universal gas constant, $c_p$ is the specific heat of the gas at constant pressure, and $p_0$ is a reference pressure.
This parameter provides a combined measure of temperature and pressure, allowing us to analyze vertical stability in a more simplified manner. By using potential temperature, we can assess how temperature and pressure variations impact atmospheric stability and the formation of meteorological phenomena.
Potential temperature plays a crucial role in meteorology and climatology, helping us understand and predict atmospheric processes and their vertical evolution. By considering potential temperature, we can examine how atmospheric stability influences cloud formation, convection, and other phenomena related to atmospheric dynamics.
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Vertical stability in the atmosphere
Equation
In the air, variations in temperature ($T$) and pressure ($p$) can cause the air column to become unstable. If we introduce the potential temperature using the equation:
| $ \theta = T \left(\displaystyle\frac{ p_0 }{ p }\right)^{ R / c_p }$ |
where $R$ is the universal gas constant, $c_p$ is the specific heat of air, and $p_0$ is the reference pressure (1000 mb).
An atmosphere with a structured variation of potential temperature ($\Delta\theta/\theta$) with height ($\Delta z$) that is always positive is stable and oscillates with a frequency given by:
| $ N = \sqrt{\displaystyle\frac{ g }{ \theta }\displaystyle\frac{ \Delta\theta }{ \Delta z }}$ |
where $g$ is the acceleration due to gravity.
If the variation of potential temperature with height is negative ($\Delta\theta/\Delta z$), the square root does not have a real solution, indicating that there is no stable solution and the layers will start to displace.
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Vertical stability in the ocean
Equation
In water, variations in density or salinity $\Delta\rho$ can make the water column unstable. When the system is stable, it oscillates at a frequency known as the Brunt-Väisälä frequency $N$, which is calculated as:
| $ N = \sqrt{-\displaystyle\frac{ g }{ \rho }\displaystyle\frac{ \Delta\rho }{ \Delta z }}$ |
where $g$ is the gravitational acceleration, $\rho$ is the density, and $\Delta z$ is the variation in depth.
If at any point the argument inside the square root becomes negative, the system becomes unstable. Therefore, we can conclude that water becomes unstable whenever the density variation becomes positive, as the expression contains a negative sign. This means that it becomes unstable when there is a higher density above a lower density, indicating that the weight causes the column to destabilize.
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Horizontal stability: Rossby number
Equation
To compare the strength of the Coriolis force with the inertial force, we can define their relationship as a dimensionless characteristic number known as the Rossby number. Since both forces depend on mass and velocity $U$, the resulting number simplifies to:
| $ R_0 =\displaystyle\frac{ U }{ f R }$ |
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 }$ |
which depends on the Coriolis factor $f$ and a characteristic length $L$.
By examining this relationship, we can see that the Rossby number represents the ratio between the characteristic velocity of the fluid and the effect of the Coriolis force. This number indicates whether the system is dominated by inertia or the Coriolis force.
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Vertical stability: critical size in air
Equation
In the limit case where the Coriolis force is of the same order as the inertial force, the Rossby number is of the order of unity. This implies that the characteristic length L is of the order of the velocity U divided by the Coriolis factor f. On the other hand, if we model the velocity using the Brunt-Väisälä frequency N and the height H, we have:
| $ \lambda_R = \displaystyle\frac{ N H }{ f }$ |
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-Väisälä 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 }$ |
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