Subfield: Thermodynamics


Question: 3.1. Consider an air parcel initially in equilibrium with its surrounding air at altitude



$z_{0}$, i.e. it has the same temperature $T\left(z_{0}\right)$ as that of the surrounding air. If the parcel is moved slightly up and down (e.g. by atmospheric turbulence), one of the three following cases may occur:

- The air parcel finds its way back to the original altitude $z_{0}$, the equilibrium of the parcel is stable. The atmosphere is said to be stable.
- The parcel keeps moving in the original direction, the equilibrium of the parcel is unstable. The atmosphere is unstable.
- The air parcel remains at its new position, the equilibrium of the parcel is indifferent. The atmosphere is said to be neutral.

What is the condition on $\Lambda$ for the atmosphere to be stable, unstable or neutral?

Solution:
In order to know the stability of atmosphere, we can study the stability of the equilibrium of an air parcel in this atmosphere.

At the altitude $z_{0}$, where $T_{\text {parcel }}\left(z_{0}\right)=T\left(z_{0}\right)$, the air parcel is in equilibrium. Indeed, in this case the specific mass $\rho$ of air in the parcel equals $\rho^{\prime}$ - that of the surrounding air in the atmosphere. Therefore, the buoyant force of the surrounding air on the parcel equals the weight of the parcel. The resultant of these two forces is zero.

Remember that the temperature of the air parcel $T_{\text {parcel }}(z)$ is given by (7), in which we can assume approximately $G=\Gamma$ at any altitude $z$ near $z=z_{0}$.

Now, consider the stability of the air parcel equilibrium:

Suppose that the air parcel is lifted into a higher position, at the altitude $z_{0}+d$ (with $d>0$ ), $\quad T_{\text {parcel }}\left(z_{0}+d\right)=T_{\text {parcel }}\left(z_{0}\right)-\Gamma d$ and $T\left(z_{0}+d\right)=T\left(z_{0}\right)-\Lambda d$.

- In the case the atmosphere has temperature lapse rate $\Lambda>\Gamma$, we have $T_{\text {parcel }}\left(z_{0}+d\right)>T\left(z_{0}+d\right)$, then $\rho<\rho^{\prime}$. The buoyant force is then larger than the air parcel weight, their resultant is oriented upward and tends to push the parcel away from the equilibrium position.

Conversely, if the air parcel is lowered to the altitude $z_{0}-d(d>0)$, $T_{\text {parcel }}\left(z_{0}-d\right)<T\left(z_{0}-d\right)$ and then $\rho>\rho^{\prime}$.

The buoyant force is then smaller than the air parcel weight; their resultant is oriented downward and tends to push the parcel away from the equilibrium position (see Figure 1)

So the equilibrium of the parcel is unstable, and we found that: An atmosphere with a temperature lapse rate $\Lambda>\Gamma$ is unstable.

- In an atmosphere with temperature lapse rate $\Lambda<\Gamma$, if the air parcel is lifted to a higher position, at altitude $z_{0}+d$ (with $d>0$ ), $\quad T_{\text {parcel }}\left(z_{0}+d\right)<T\left(z_{0}+d\right)$, then



$\rho>\rho^{\prime}$. The buoyant force is then smaller than the air parcel weight, their resultant is oriented downward and tends to push the parcel back to the equilibrium position.

Conversely, if the air parcel is lowered to altitude $z_{0}-d(d>0)$, $T_{\text {parcel }}\left(\mathrm{z}_{0}-d\right)>T\left(\mathrm{z}_{0}-d\right)$ and then $\rho<\rho^{\prime}$. The buoyant force is then larger than the air parcel weight, their resultant is oriented upward and tends to push the parcel also back to the equilibrium position (see Figure 2).

So the equilibrium of the parcel is stable, and we found that: An atmosphere with a temperature lapse rate $\Lambda<\Gamma$ is stable.

![](https://cdn.mathpix.com/cropped/2023_12_21_cb6ddd00156af4e446fbg-1.jpg?height=648&width=691&top_left_y=981&top_left_x=425)

$$
\begin{aligned}
& T_{\text {parcel }}>T \Rightarrow \rho_{\text {parcel }}<\rho \quad \text { up } \uparrow \\
& T_{\text {parcel }}<T \Rightarrow \rho_{\text {parcel }}>\rho \quad \text { down } \downarrow
\end{aligned}
$$

Figure 1

![](https://cdn.mathpix.com/cropped/2023_12_21_cb6ddd00156af4e446fbg-1.jpg?height=654&width=699&top_left_y=1803&top_left_x=427)

$$
\begin{aligned}
& T_{\text {parcel }}<T \Rightarrow \rho_{\text {parcel }}>\rho \text { down } \downarrow \\
& T_{\text {parcel }}>T \Rightarrow \rho_{\text {parcel }}<\rho \quad \text { up } \uparrow
\end{aligned}
$$

stable

Figure 2



- In an atmosphere with lapse rate $\Lambda=\Gamma$, if the parcel is brought from equilibrium position and put in any other position, it will stay there, the equilibrium is indifferent. $A n$ atmosphere with a temperature lapse rate $\Lambda=\Gamma$ is neutral

Final answer: $\Lambda>\Gamma$ is unstable
$\Lambda<\Gamma$ is stable
$\Lambda=\Gamma$ is neutral