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Simple Mid-Air Vanilla Physics

By
Person \(\pi\)

Date : 03 Jul 2020, Friday
Location : 11580 metres above Dong Hoi, Vietnam

Dear reader,

I left Hong Kong (and LPCUWC this morning) on a government organised relief flight. It was bittersweet. I will make sure I will see these people again somewhere in future, wherever they are in the world. I owe them everything for my welfare and growth for the past two years.

Now, I am 11,580 metres above the Vietnam grounds and I just woke up from my sleep. I can see this mesmerising view out of my window and decided to do a brain exercise.

Disclaimer: From here on, it is going to get nerdy and will contain some excessive appreciation for Physics, Maths and Philosophy of Science.

In computer programming, there is something called 'Vanilla code'. It means that you write everything from scratch without relying on any pre-existing code written by others. It is inefficient, slow and painful, but trains your brain well on computational thinking.

When I study anything, I always approach from both sides. In one end, I love exploring the ideas and frameworks constructed by others, appreciate them and stand on the shoulders of the giants. However, also on the other end, I need to (ideally) make sure whether I can build these ideas again from scratch without referencing anything - For example, will I be able to build a simple sundial to keep track of time when I am stuck alone on an island? Am I able to propose an explanation on how an analog electronic gadget works by looking at it for a few minutes? (I say analog because most digital devices are black boxes these days.)

Of course, these are ridiculous questions to ask when you are having noodles together with your friends or family, or when you are running to catch the last bus available. And of course, the ability to have food to eat and time to think about these is a privilege. However, this kind of thinking is how you get really intimate with Physics or maths or computers or Asian languages or painting or music or just about anything you study, and appreciate their true beauty for the sake of it, in peace and tranquillity.

Enough love talk. Back to business.

I have never studied clouds (Both the real clouds and the cloud on computer), or meteorology. They were never a part all of the high school physics curriculums I have experienced.

And now I am in mid-air with no access to WiFi, my phone is on airplane mode (so I cannot ask Mr. Google). However, I am going to use very basic high school physics to explain why I should be seeing the clouds like that from my airplane window, without reading anything about clouds at all. I will check whether my explanation is correct or not later. In other words, I am going to do vanilla Physics.

Observations

Now, I can see that there are at least three types of clouds. I do not know their proper names.

They are mainly -

My observation is that the big lumpy ones are almost always below the small, thin flat ones. And hazy ones have the same height as the thin flat ones.

What are clouds? Clouds are water vapour in air (Technically water which is vaporised but condenses in mid-air. However, when they condense, the size of the droplets is so small such that effect of gravity on them is negligible. Thus, they can stay in mid-air without falling. When they are large enough to fall, they become rain.) Since, clouds have water in them, they will have the mass of water and weight due to gravity. If they stay on mid air alone, they should be dropping down.

But why are they remaining on mid air? In Statics, for an object to remain at equilibrium, there must be no net force on it. Therefore, the downward weight of the water vapour in cloud must be balanced by an upward force of equal magnitude to remain at a certain height. (It is okay to have the wind blow from the side and provide a force from the side, that will be how clouds move sideways)

What is providing that upward force? We all learnt that when air/ or any fluid is heated (by the sun, in the case for clouds), it expands. Expansion of air makes its density lower. Due to Archimedes Principle, it will occupy more space than air at normal temperature and thus experience greater upthrust. (I am treating air as a fluid in this case). Therefore these hot air rise upwards and collide with the clouds, thus exerting an upward force on the clouds (or the very small water droplets inside them.) Currently, I am making a hypothesis that these upward rising hot air stream is giving this upward force on the cloud to balance its weight.

My Question

Yes, we know clouds are going to hang in mid air since effect of gravity is negligible on the water droplets. However, why do different clouds have different heights?. For instance, if two regions of water vapour condenses in mid air at a position, one into a small cloud and one into a big cloud, why does bigger one should become stable at a lower height? We will be using some maths to determine why there should be different layers of clouds.

Oversimplified Assumptions

I am assuming the following for my back-of-the-envelope calculations.

Back-of-the-envelope calculation

First of all, we have clouds with mass mc, and their weight will be Wc = mc . g (c stands for cloud).

And we have hot air rising from the ground upwards due to expansion, because of the heat from the sun. I am going to treat this rising hot air as bubbles of hot air in atmosphere (just like bubbles in water) - so that we can visualize it.

The upthrust on these hot air bubbles can be determined with Archimedes principle, which states that the upthrust force , Fu is equal to the weight of fluid displaced. Here, volume V is the volume of hot air bubble.

Assuming that these hot air bubbles are fluid, their volume can be determined by the ideal gas equation (which is of course, an over simplification).

From here, we know that the volume of the air bubble is a function of temperature and pressure.

Now we know that the temperature of these air bubbles is going to be increasing due to heat from sun (They will eventually cool down as they reach a greater height, but for now, let's assume that the heat from the sun which goes into the air bubble is not lost - or at least not lost just before hitting the clouds).

Also, we know that atmospheric pressure decreases with increasing height (as the hot air bubble rises). Thus, the volume will still be expanding. This is just like how air bubbles in water in water gets bigger as they move closer to the surface of the water.

Therefore, we know that the volume of these hot air bubbles can be written as an increasing function of temperature and atmospheric pressure.

So, we know the upthrust on these air bubbles will be increasing.

Effect of Drag Forces

However, drag forces are going to slow these hot air bubbles down, right? Well, we also know that air is the most dense near the ground and air density decreases with increasing height. What it means is that high above, there are fewer air molecules than there are near the ground (That's why it's hard to breathe at higher altitudes, for example on very high mountains.) Less dense air means drag force is not going to increase that much and we can ignore them.

Therefore, we can now anticipate that these hot air bubbles are going to rise up initially with an accelerating speed, until the air becomes less and less dense so that the upthrust begins decreasing. This will happen when density of air, ρa becomes very small to overcome the increasing function f(T,P).

However, let's assume that this is not happening at the time when these hot air bubbles hit the clouds (yes, too much assumptions, that's why it's called back-of-the-envelope calculation).

When these hot air bubbles hit the clouds, they will exert an upward force equal to the upthrust onto the clouds. Many of such hot air bubbles can hit one large cloud.

Let's say we have two clouds with different volumes for now.

At equilibrium, the sum of upthrust forces from the air bubbles will balance the weight of the cloud.

This can be described mathematically as (Summation is here because there are more than one hot air bubbles hitting the cloud) :

The equation above (Net Force = zero) is the equilibrim condition. The clouds will stop falling down when this condition is met.

Solving the different heights problem

However, let's take the case when two regions of water vapour in air begin to condense into clouds at the same height. One is going to condense into a big lumpy cloud and another region of water vapour will evolve into a very small flat cloud.

Of course, the big lumpy one has bigger mass and weight. This weight will cause both clouds to fall down. However, they will be accelerate downwards due to gravity first, but decelerate later due to the upthrust force form the rising hot air bubbles from downwards.

Thus, their downward acceleration can be calculated from the net force (assuming constant acceleration).

We know from the Second Law that Net force with constant downward acceleration is Fnet = ma.

We can rearrange the mass of cloud. And mass of cloud is density times volume.

This is the downward acceleration formula for the cloud for some duration after formation, before the f(T,P) increases large enough so that this downward acceleration becomes zero and the cloud becomes stable at a height.

Currently the equation takes the form a = constant - constant/Volume of cloud. Therefore, a quick sketch of the graph should look like the following (Why? Because 1/x is a reciprocal function. The negative sign flips it vertically. Adding one (or a constant) shifts it upwards vertically).

Thus, from analyzing the graph, we can see the full story on why the bigger clouds should become stable at lower heights.

The bigger cloud with larger volume will have greater downward initial acceleration for a while. Therefore, they will reach to the bottom and become stable compared to smaller, flat clouds with less volume.

Just before landing. See? Big lumpy ones at bottom, flat, thin ones at the top.

Of course, there are exceptions to everything. Here is a big lumpy one which made it to the height my airplane is at. (I suspect that upward convection current of air here is a lot stronger to make this guy stable at this height - in other words, it’s hotter in this region )

How about the big hazy ones?

Maybe they have non-uniform density? Or water condenses at different heights and merge into one big hazy one? I am not sure and have not taken them into account here.

Fact Check

Now I have a chance to go to wikipedia for a quick look at cloud types. Not bad. The big ones I think may be Cumulus and Stratocumulus. There is Stratus at the same level with them.

Photo from Wikipedia

However, smaller ones such as Altocumulus, Alsostratus, Cirrus, Cirrostratus should be at the top. This agrees with our hypothesis.

And there is a big hazy one Cumulonimbus, which I have no idea how it forms.

And the variation of temperature with the altitude is quite interesting. I was not expecting that.

[Image Source]

My model looks like a kid's toy here. It only explains why bigger clouds should be mostly at the bottom rather than on the top (not taking the exceptional cases). However, it was fun constructing it.


Sincerely,
\(\pi\)


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Oh, Hi ! Hokusai
24 Jul 2020, Friday
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The Coding Samurai
03 Jul 2020, Friday
Downtown Yangon, Myanmar

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