Saturday, 9 May 2020

How to work out the 'adiabatic lapse rate' using logic and maths (instead of relying on observations).

Some of my posts drag on a bit before the conclusion, so this time I will do bullet points first (with links to skip down to the fuller explanation).

1) The traditional explanation, that the sun warms the Earth's surface; the surface warms the air above it is largely nonsense, or is only a small part of the picture, if you bother thinking about it for a few minutes. Fuller explanation.

2) The actual explanation is that energy tries to spread out and equalise in all directions. The air at the surface has no potential energy but lots of heat. Further up, the air has more potential energy but less heat, all the way up to the top of the troposphere. The two forms of energy (potential plus heat per unit volume) is a constant all the way up. Fuller explanation.

3) Some constants and formulae we have to accept as a given; we plug these into the workings; and work backwards to get the lapse rate that balances Joules/m3 all the way up. Constants, formulae and workings.

4) Conclusion: to get the Joules/m3 to balance all the way up (I calculated up to 6km, half-way up), the local temperatures of the troposphere has to fall by 8C (or 8K) for every 1 km you go up. The extra potential energy = the energy no longer needed to warm the air in each m3.

That's a bit more than the observed lapse rate of 6.5C, and a bit less than the 9.8C which Wiki says is the theoretical dry lapse rate. So I'm pretty happy with it for now.

5) How to set up hyperlinks within one web page, from

1) The traditional explanation is (largely) nonsense.

You're seven years old and your science teacher tells you that the Sun warms the earth surface; this warms the air above it; that warms the air above that etc. So the bottom layer is warmest and it gets colder the higher you go. So kids, she trills, that's why it's so cold at the top of Mt Everest.

Sounds plausible until you relate it to real life...

- there is a lapse rate at night, when the surface is not being warmed and it's actually cooling down faster than the air. Which is why you should have more layers underneath you than above you when sleeping outside.

- there is a lapse rate, even when it's cloudy and the surface is not being warmed; the sun is hitting the clouds first. It's still colder above the clouds than below.

- the Sun (when it's shining) hits the top of Mt Everest with the same intensity as it hits land at sea level; but it doesn't warm it up nearly as much.

- the surface cools down from average 293K by day to average 273K at night (let's say for sake of example) and warms to 293 again the next day. So 3.5% of the heat is lost and regained each day, it has a 'half life' of about three weeks. Most of the heat is left over from previous days and that has had plenty of time to mix evenly (if it wanted to do so).

- it's like saying "Bath water is deeper at the tap end, because that's where the water goes in." Wrong, the surface quickly levels off. Bath water is indeed deeper at the tap end, but it's because that's where the plug hole is, and the plug hole end is lower.

- Yes, on a sunny day, you can see hot air rising from e.g. dark tarmac = 'thermals'. Vultures and hang glider pilots use these to get around. But this is a dynamic thing, it disturbs the equilibrium, which is constantly trying to re-establish itself, thermals are not the equilibrium or an explanation for it.

2) The actual explanation.

- I hope we all remember "Energy cannot be created or destroyed, it just transforms from one from to another". The sun sends us energy as light; plants use it to grow and they store chemical energy. You burn wood (stored chemical energy) in a power station and convert it to heat; that turns water to steam; steam gets converted to kinetic energy and water (again); the kinetic energy gets converted via magnetic to electrical energy. Which goes down the wires, through a filament in a bulb and becomes light energy and heat; the same as what we got from sun a long time ago.

- energy will choose whatever form (radiation/light, magnetism, electrical, heat, sound etc) enables it to spread out as evenly and as quickly as possible for a given medium.

- heat will flow from warm to cold (speed depends on how good the medium insulates); photons will go from light to dark (which is why it takes them so long to leave the sun; it's equally bright everywhere and they're not sure which way to go); once the photons have escaped the sun, they whizz off at the speed of light; sound waves travel as fast as they can from loud to quiet (760 mph through the air at sea level; faster through solids).

- gases, i.e. the atmosphere, are very compliant. Energy can travel through it by conduction; by radiation; or by convection i.e. converting heat to potential energy (air will carry sound and shock waves, and even conduct electricity if the potential difference is high enough = lightning). The precise method and speed are immaterial in an equilibrium situation. They all boil down to the same thing and the speed is to all intents and purposes instantaneous. If heat from the atmosphere hits the edge of the atmosphere, it doesn't care, it turns into light/radiation and keeps going, the only way for energy to travel through a vacuum.

- on a more mundane level, a hot air balloon is a way of converting heat energy to potential energy (altitude). If it were 100% efficient, you would gain exactly the same amount of Joules in potential energy as you use up in Joules of heat energy. The troposphere is as close to 100% efficient as you can get.

- so it seems reasonable to assume that the total energy per unit volume (like a m3) will be the same at all altitudes in the troposphere in an (idealised) equilibrium situation, let's call this 'energy equivalence' for the sake of calling it something.

3) The constants, formulae and workings.

The main steps:

1. Set up your spreadsheets with all the scientific constants and formulae.

2. Find out surface/sea level temperature and plug that into Barry to find the altitude at which pressure = 0.5 atm, which is 5.9 km.

3. Take 0.5^(1/5.9) to find the % by which pressure falls for each km altitude.

4. Work out total Joules/m3 at sea level.

5. By trial and error, find the temperatures/lapse rate which give you the closest amount of Joules/m3 all the way up. You get the best fit with 8K per 1,000 metres.

So the result of Barry feeds into this calculation and vice versa. Doesn't matter, it is not a circular calculation if there is an equilibrium and the results match up to observations.

Bottom row shows I'm pretty close up to 5 km and only 3% out at 6 km altitude. The measured temperature half-way up is apparently 250K, not 240K. But overall, I call it a good result for a couple of hours with a calculator in the back garden. Click to enlarge.


mombers said...

When are you submitting this to a journal?

Mark Wadsworth said...

M, thanks. I have merely re-invented the wheel and described it in my usual folksy way.

I think this is what the clever formula on Wiki means, but that is impenetrable to me so I started again from scratch.

Stephen Stretton said...

Pretty good.

Starting point. Physics 101: there are THREE process of heat transfer (Conduction, Convection and Radiation), two of which are important here (forget Conduction).

The the Earth as a whole interacts with the rest of the Universe by only ONE of those processes, Radiation. That's how the Earth gets heat from the universe (ok, 99.99999999999999999999% of it comes from the Sun, starlight can be discounted). And it's also how it gives off the energy to the universe, by Black body Radiation. Which is a bit of a misnomer in the sense that a good example of black body radiation is putting a lump of iron into a furnace and seing it GLOW red. Our planet Glows, but it glows Infra red because it's not hot enough to glow red. You see light is a form of 'electromagnetic radiation' and so is infrared - it's a spectrum, so to speak ;) (Indidentally some molecules with three atoms -- basically they are like sets of springs, and to 'bend' has a certain resonent frequency -- CO2, H20 absorb infrared radiation, and THEY then reradiate, but some of it comes back to earth hence the standard greenhouse effect)

Anyway back to your question. The key thing is that the bottom of the atmosphere (ie basically the atmosphere, the stuff higher up is so thin that it's really just a bunch of atoms flying around not interacting very much) equilibriates by CONVECTION. How this works exactly i'm not an expert in, but I understand it's helpful to imagine a particular process called an ADIABATIC one . An adiabatic process does NOT transfer heat but it does do WORK (or have WORK done on it). So imagine a parcel of air. So for example a baloon. Now a parcel of air would change volume as it rises, so the volume goes up and the pressure goes down. (Actual rising is besides the point, it could just as well be falling -- the point is we are linking what a parcel of air would be like at ground level to what it would be like a bit further up). In that adiabatic process, the internal energy is not QUITE conserved. As a body of air expands it does some work on the surroundings - pushes the air a bit father up than it would otherwise be. That happens due to the lower pressure. So you can see that there's this pressure-density relation, and then there's the work that this change in density of this parcel of air does on the surroundings which leads to the CHANGE in internal energy. So I think you are almost right in your suppositions here (although I'm known for being charitable) but you should also account for the work done when a parcel of air changes notionally from being small (at high density) to being large (at lower density).

Dinero said...

> St Strett.

An analysis based on a location relative to ground level doesn't work because temperature gradients also exist horizontally for volumes of gases adjacent to a heat source.

Stephen Stretton said...

> An analysis based on a location relative to ground level doesn't work because temperature gradients also exist horizontally for volumes of gases adjacent to a heat source.

Erm, 'doesn't work'??? That's a different phenomenon. I'm not trying to explain the universe with a single story. There's lots of thing that my explaination doesn't explain. Look into convection and how it works horizontally if you wish. Probably the answer is those circular flows that you'd have in a room with a radiator.

Mark Wadsworth said...

SS, thanks, I have replied by email.

Ignore Dinero, he's a troll and won't accept any ground rules of sensible discussion.

Dinero said...
This comment has been removed by the author.
Dinero said...

I am not doing " a troll " You put forward an idea and I point out what is wrong with it. If you don't want any discussion on a scientific post then say so , no problem.

If you do want a discussion then it is a valid point of discussion. Scientific descriptions that aim to be encompassing can not be on a case by case basis.

A description of a vertical lapse rate can not ignore that it also appears without gravity, horizontally.

Mark Wadsworth said...

Din, OK. I'll try again...

May I remind you what I am explaining here - the vertical lapse rate. Not how heat from a hot object spreads horizontally (most of it goes vertically, as it happens). Those are the ground rules of this discussion.

If you think I have made a mistake in my logic or calculations re the *vertical distribution of energy*, I'd be interested to hear it.