# Of potential temperature and equivalent potential temperature

So my supervisors did their supervisor thing and highlighted some holes in my knowledge, so I’m going to fix that here!

Lets get some definitions out of the way $\theta$ (theta) is potential temperature and $\theta_e$ is equivalent potential temperature.

What do they mean and what could they tell us about the atmosphere?

Potential Temperature

Remembering that potential energy is the energy that an object might have depending on its position (so a ball that is held up in the air has more potential energy than the one on the floor) could the potential here be a bit similar?

It is sort of!

Potential temperature can be defined as:

The temperature a parcel of air at pressure P would have if adiabatically brought to a standard reference pressure (normally 1000hPa).

Hmm, what does adiabatically mean again?

An adiabatic process is one which occurs without exchanging heat with its environment. Unlike a diabatic process, which is the opposite.

As I like equations, we can write one for potential temperature,

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

where $T$ is temperature in K, $R$ is the gas constant and $c_p$ is the specific heat capacity at a constant pressure.

We could use potential temperature to determine if the atmosphere is stable or unstable. If we were to measure the normal temperature as we walked up a mountain, it would most likely get colder as we got higher, maybe demonstrating an unstable atmosphere. Or maybe not, this is mostly due to the pressure change. Potential temperature can show if we really have stable or unstable conditions.

Normally potential temperature will increase upwards with height, and it is conserved (doesn’t change) for all dry adiabatic processes. It would change if we have heating, cooling, evaporation or condensation.

So how do we tell stability? Well if potential temperature increases with height than the atmosphere is statically stable. If it decreases with height then the atmosphere is statically unstable and convection is likely.

[Aside: So why does that mean it is stable?

If we put warm, less dense air under cold dense air, we’d expect it to rise up and swap over with the cold air. That’s because it is unstable. Now in the atmosphere, this is made more complicated by the fact the air expands as it rises as the pressure is lower (so its not being squished in as much and can be freeeeee…) when this happens it gets colder as it uses up some of its energy to expand. Now going back to our mountain, the warm air at the foot of the mountain could rise up all it liked, but it would be colder than the surrounding air by the time it got to top of the mountain. Then it would sink back down, so the change (perturbation) has corrected itself. This makes the atmosphere stable!

If it had a higher potential temperature it would still be warmer by the top of the mountain and would thus actually be unstable. ]

For me, I’m trying to see if I have data showing a change from stable to unstable atmosphere, so this is super useful! Here’s a plot!

With high values in red this looks pretty stable don’t you think? Those black lines are where the plane was flying when we were recording data.

How about equivalent potential temperature?

This is slightly more complicated, as it takes into account the amount of water vapour in the air parcel.

The definition of equivalent pot. Temp. is the temperature a parcel of air would have if all the water vapour in the air were to condense, releasing its latent heat, and the parcel was taken adiabatically to a standard pressure (1000hPa).

The equation for this isn’t so nice, in fact it only has approximate equations, but they will have to do. You can find them on the wikipedia page. (I’m too lazy to type them…) (http://en.wikipedia.org/wiki/Equivalent_potential_temperature)

Using equivalent potential temperature, we can determine stability in saturated air.

But why would we have a problem with saturated air?

Like with normal temperature above, sometimes potential temperature can make the atmosphere look stable when it isn’t. Remembering that condensation releases heat to its surroundings, if a parcel of saturated air moves upwards the vapour is going to start condensing, so warming the air parcel, so it might still be warmer than the surrounding air when it gets to the top of our mountain. Thus being unstable.

Again the atmosphere is unstable if $\theta_e$ decreases with height and stable if it increases with height.

This is also useful for me as I’m looking at convection over the sea! Look at another plot!

This is for the same flight, looks pretty similar so probably the water vapour isn’t causing an issue here. Though my supervisors are confused about the units (as am I).

Hurrah for learning!