- Define the potential temperature for a fluid

The potential temperature is the temperature a parcel of fluid would achieve if it were transported adiabatically to the Earth's surface (or some reference pressure).

- Express the potential temperature of an ideal gas as a function of its temperature and pressure.

The formula is as follows:

$$ \theta = T \left( \frac{P_0}{P}\right)^{\frac{R}{c_p}} $$

where T is temperature, P is pressure, R is the ideal gas constant, and \(c_p\) is the specific heat capacity of dry air at constant pressure.

- Derive the adiabatic lapse rate \(\Gamma_{\text{ad}}\) for an ideal gas (in hydrostatic balance).

The lapse rate is the rate at which the temperature changes as one ascends through the atmosphere. Since we're talking about the **adiabatic** lapse rate, it's the rate the temperature changes along a dry adiabat, which means that the potential temperature is conserved along this path. So

\begin{align*}D\theta &= D\left[T \left( \frac{P_0}{P}\right)^{\frac{R}{c_p}} \right] \\ &= 0 \end{align*}

This will give us a relationship between temperature and pressure changes:

\begin{align*}\frac{\theta}{T}DT &= \frac{R}{c_p} DP\frac{\theta}{P}\end{align*}

And this leaves us with

$$ \frac{DP}{DT} =\frac{c_p}{R} \frac{P}{T} $$

Assuming a steady state scenario with variation only in the vertical, we can discard the total derivative for the partial derivative with respect to \(z\). Letting \(\Gamma = - \frac{\partial T}{\partial z}\), we have

$$ -\frac{\partial P}{\partial z} = \Gamma\frac{c_p}{R}\frac{P}{T}$$

We only need two more equations: (1) hydrostatic balance and (2) the ideal gas law

$$ \frac{\partial P}{\partial z} = -\rho g $$

$$ P = \rho R T $$

Substituting both of these, we get

\begin{align*}\rho g &=\Gamma\frac{c_p}{R}\frac{\rho R T}{T} \\ \implies \Gamma &= \frac{g}{c_p}\end{align*}

- Discuss the behaviour of the fluid when the actual lapse rate \(\Gamma > \Gamma_{\text{ad}}\) and \(\Gamma < \Gamma_{\text{ad}} \).

This is a question of whether our parcel cools down more drastically with height than the environment, or vice versa. Remember, we define the lapse rate to be defined negatively with respect to the change in temperature with height. So the more positive this is, the more drastically a parcel **cools** with height:

- If \(\Gamma > \Gamma_{\text{ad}}\), then as our parcel rises adiabatically, it finds itself cooler than the rest of the atmosphere, thus being restored back to its place. This is a
**stable** scenario.
- If \(\Gamma < \Gamma_{\text{ad}}\), then our parcel becomes warmer than the rest of the environment as it rises, leading to rise further. This is an
**unstable** scenario.