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So to understand how convection sets up this temperature difference between

the ground and the upper atmosphere that drives the greenhouse effect,

we need to talk about actually three or four separate pieces of physics.

That may seem kind of unrelated and

off the wall to you, but we'll put them all together when we get there.

So the first is about pressure in a standing fluid.

So as we stand here in the atmosphere or

if we were to stand at the bottom of swimming pool, we feel pressure.

And the pressure that we feel is due to the weight of the material over our heads,

sort of pushing down.

So standing at the bottom of the swimming pool we've got this water over our head

and that, the gravitational attraction of the water is what we feel as pressure,

and the same is true of the atmosphere, but the difference between the water In

the air is that water is basically incompressible.

So it's basically all the same density all the way up to the top whereas

gasses expand if they're not under pressure.

And so, the gas is much thicker close to the ground, and

then it gets thinner as you go up in the atmosphere.

So what that means for pressure in a incompressible medium like water,

if you start at the surface of a swimming pool and you swim down,

the pressure is going to increase linearly with the depth.

So the pressure would be equal to some pressure at the top plus a constant

times the the depth, and so this describes the straight line.

And that's because if you start at the top and

you swim down a meter you've got one meters worth of water over your head, and

you do it at all the way at the bottom of the pool, one more meter.

The water has the same density at the top as at the bottom, so

the slope of this line is constant, meaning it's a straight line.

Whereas, in a compressible medium like a column of air,

the pressure change if you climb up a little ways in the atmosphere

is more intense than if you're already at the top of a mountain and you go up

a little bit higher, because the air is so much thinner up at the top of a mountain.

So if you climb up say, 100 meters at the top of a mountain,

you don't leave behind as much air as if you were at the ground or sea level and

you climb up 100 meters, you're leaving behind more air.

And so, that means that the pressure isn't linear with altitude like it is in water,

but it sort of gets more and more gradual as you go up.

It follows an exponential profile.

So the pressure at some height is equal to the pressure

at the ground times e which is a number raised to the power

of the height divided by a value of about eight kilometers.

So one thing to note about this atmospheric pressure profile,

is it never actually really reaches zero.

Mathematically, it gets closer and closer and closer, but it's as if the pressure

of the Earth's atmosphere extends out forever into the entire galaxy.

We just kind of, an idea that only a mathematician could love, probably.

But, it might give you the impression that the atmosphere is infinitely thick.

But actually, this multiplier here in the exponent is an important number, and it

sort of sets the height of the atmosphere in some way is called the scale height.

So that number is about eight kilometers.

And it turns out that if you could make

the air all have the same density has air at the ground.

So that, it was like incompressible like

water that the atmosphere would run out at eight kilometers high.

So eight kilometers is kind of like how thick the atmosphere is in some sense,

even though mathematically it is infinitely thick.

But, this tells you a scale height, so

when you're actually at 8 kilometers the pressure is a factor of

1 over e times this pressure which is about 30% or something like that.

It sort of sets the height scale of the atmosphere.

So the next thing that we need to worry about is what happens

to the heat when this gas expands as it rises up, or

when it gets compressed when it comes down toward the earth under higher pressure,

because it affects the temperature of the gas actually.

And you can think of this by imagining

a piston that's got a sleeve that's insulated here, so

that no heat can cross the walls, and then you've got this

plunger thing that you can sort of push in there to compress the gas.

So you push on this plunger and

you're actually doing work to do that, you're pushing the gas molecules together.

It turns out that where that energy goes from doing the work,

is to make the temperature of those gas molecules higher.

So you may have encountered this if you've ever gone to a dive shop

to go scuba diving.

You take a scuba tank and you need to get your tank filled up.

And so, they could just pump air into it.

But when they do that, it gets very hot, and

the tank doesn't wanna hold more than a certain amount of pressure.

So hot gas is exerting more pressure, and so if you went to a cheap dive shop, and

they just put air in your tank, and let it get hot, you would then take the tank and

go jump in the cold water, and the pressure would go way down.

You'd only have like half a tank of air to breathe.

So a good dive shop will put the scuba tank into a big tub of water, and

cool it down as you put the gas in to prevent

the heating up of the gas from effecting how much gas you take.

Or you can feel it if you've ever let the air out of a bicycle tire.

You can feel the expansion of the gas as it's coming out of the tire

makes it get really cold.

You can feel it on your thumbnail when you do that.

The air when it was inside the tire,

was at the same temperature as the air outside.

But just because it expands, the fact of its expanding decreases

the average kinetic energy of the molecules of gas, so it makes it colder.

[NOISE] Convection is the process of

heating a fluid from below and

causing it to overturn, or

cooling it from above.

If you heat from below, you make the fluid at the bottom expand, because it's warmer.

And when it expands, it's less dense and it will tend to rise up.

Or if you cool if from above, it will contract, and that makes it more dense and

it will tend to fall down.

So if you were to, this is a picture of a lava lamp.

You've got to have the lightbulb at the bottom to heat it from below.

If you had a lightbulb at the top it wouldn't convect,

it wouldn't be any fun, it wouldn't work.

So this happens in the real world when sunlight hits the ground.

It warms the atmosphere from below and that causes the atmosphere to convect.

Or the ocean can convect in a sort of upside down way,

if you cool it from the top, like around Antarctica or

up in the North Atlantic, you make the water at the top denser,

and so it sinks and you do this over turning thing.

[SOUND] So it's easiest to understand convection

first in the incompressible case, because its just sort of simpler.

We'll start from a case where, let's say, we've got a pan of water on the stove and

we're heating it from below, so it's gonna be convecting.

So let's start out and say that the water is well-mixed in the pan, and

so that means that it will all be the same temperature.

Because there isn't any expansion of the gas that makes it

change its temperature or anything, because gases, of the liquid, I mean.

Because liquids are not compressible.

So it's all the same temperature.

Just like if you put some sugar into your coffee and stir it,

it would all have the same amount of sugar all through the column.

So then, we heat it from below.

So if this is the temperature we make a little blob of hot water at the bottom,

but that's gonna expand and it's gonna be less dense than the water above it,

and that's not the way it wants to be, so it's gonna rise up.

Now, in the lava lamp, they have two different kinds of fluid that can't mix.

It's like oil and water.

So the blob from the bottom gets hot and it rises.

It goes all the way to the top,

and then it cools down and comes all the way to the bottom again.

But what tends to happen in the atmosphere is that when this blob of hot stuff rises,

it kind of mixes with everything above it, and so

it tends to move the whole temperature of the whole column to a higher

value, instead of just this blob going up to the top like it would in a lava lamp.

Now, in a compressible column of fluid of gas,

we start out with a well-mixed profile, but now, here's the tricky part.

A well-mixed profile is not all the same temperature,

because this gas up here is under lower pressure.

If you were to take this gas, and

pull it down to the bottom of the column so that it was under higher pressure,

that would make it warm up, because it's being compressed.

And if this is a well-mixed profile,

you would just exactly follow this trajectory going down.

So a well-mixed profile of gas is not all the same temperature,

because of this expansion effect.

And then, the rest of it is pretty easy.

We heat it from below and mix it, and

it goes to another well-mixed profile which is parallel to the first.

Pretty much but warmed up, but

the whole thing is kind of just tilted over by this expansion effect of the gas.

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Pressure in a Standing Fluid

Global Warming I: The Science and Modeling of Climate Change

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