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Â So our first climate model was too cold and the remedy to that is to add

Â the greenhouse effect and we're gonna do that in a way that's very, very simple.

Â You wouldn't understand the greenhouse effect without understanding this

Â simple model that I'm going to show you.

Â But of course, reality is much more complicated.

Â So after we put together this very simple model and understand it,

Â we'll be spending a fair bit of time in the class actually talking about how to

Â make it more realistic to make it more like the real world.

Â So the bare Earth model that we looked at last time was basically this.

Â It had sunlight coming in and energy leaving according to epsilon,

Â sigma, temperature of the ground to the 4th power.

Â So our new model with the greenhouse effect, we're gonna suspend a pane

Â of glass above the ground and the glass has the property that sunlight can

Â go through it without any impediment, it's not absorbed by the glass.

Â But any infrared that comes from the ground is gonna be absorbed by

Â that pane of glass and the glass itself is going to shine in the infrared

Â in both directions up and down according to its own temperature.

Â The temperature of the pane of glass or as I've written the atmosphere,

Â the temperature of the atmosphere here.

Â So we're assuming throughout all of this that these epsilon values

Â are equal to one, so that means that the glass has all of

Â the oscillators that it needs to make a nice smooth black body curve and

Â create the full sigma T to the 4th black body spectrum.

Â Of course, it's not a perfect black body.

Â Because if it were, it would absorb the sunlight too.

Â According to the definition, it should absorb and emit all frequencies.

Â So it's kind of a selective, infrared kind of a black body.

Â And we're gonna solve for the equilibrium temperature such that

Â the energy is in steady state, just like we did before.

Â [SOUND] We're gonna look for the energy in and equate that to the energy going

Â out and there are multiple places where we can construct a budget in this new model,

Â because there's actually multiple temperatures.

Â There's a temperature of the ground and the temperature of the atmosphere.

Â So starting from the ground, since we are sort of ground based

Â beings after all, we can just look at the arrows going into and

Â coming out on the ground and write them as a inputs and outputs.

Â So what's coming in to the ground here, we've got the same sunlight as before and

Â we have this arrow of infrared coming down from the sky.

Â And so that's epsilon, sigma, temperature of atmosphere to the 4th power,

Â that's what's coming in.

Â And then what's going out of the ground is just sigma T,

Â temperature of the ground to the 4th power there.

Â So that's not really as convenient as the last equation we looked at,

Â because we have two unknowns in it.

Â We can't solve for both of them at the same time with just one equation.

Â If you wanna solve for two unknowns, you need two constraints, two equations.

Â So here is another one.

Â This is the budget for the pane of glass.

Â So we have what comes into the pane of glass is not the solar energy at all.

Â It is in fact, the epsilon, sigma,

Â temperature of the ground to the 4th power.

Â That's this arrow coming into the glass there and

Â then what's leaving the pane of glass is epsilon, sigma,

Â temperature of the pane of glass to the fourth power.

Â And there's this factor of two, because the light is going both upward and

Â downward.

Â This two is not the most intuitive thing, you wouldn't probably have thought of it.

Â But without that factor of two there,

Â there actually wouldn't be a green house effect as we'll see.

Â So this is now an equation that has two unknowns in it,

Â the temperature of the ground and the temperature of the atmosphere,

Â rather haphazardly written there.

Â And so we could in principle, take these two equations and

Â use algebraic substitution to solve for just one of the temperatures.

Â And then use that, plug that back in to solve for the other one.

Â And you're welcome to do that, if you like to try your algebra chops, but

Â there's an easier way to do it.

Â And it actually has a conceptual benefit,

Â as well that we can draw a different budget and

Â it is a budget for the Earth system overall.

Â So we can draw a line, sort of a boundary

Â to space above the atmosphere.

Â And the energy crossing this boundary to space coming down has to balance

Â the energy crossing the boundary to space going up.

Â In the steady state, everything has to be in balance.

Â And so if we write those equations, what's coming down

Â is just our familiar sunshine L(1-alpha) over 4.

Â And then what's going out is epsilon, sigma,

Â temperature of the atmosphere to the 4th.

Â So this equation should look familiar to you.

Â This is actually the same equation that we got from the last model for

Â the temperature of the ground.

Â Only now, it's that the temperature of the upper layer.

Â And it turns out, this is sort of a general property of these simple models

Â and also of the more complicated Earth system that the temperature

Â where the energy shines out to space is kind of a fulcrum point.

Â It's fixed by how bright the sunshine is and

Â how much is reflected away in the albedo.

Â So this is the naked Earth model and the temperature of the ground is 255 Kelvin.

Â Here is our greenhouse model with one pane of glass here and

Â the temperature of the pane of glass is 255 Kelvin again.

Â In the exercises,

Â you're gonna work out the balance of this super greenhouse model.

Â It has two panes of glass, so twice as much greenhouse forcing as that and

Â what you will find is that the top layer is 255 Kelvin.

Â And you're also work out a nuclear winter scenario where there's cred in

Â the atmosphere and so the sunlight gets absorb in the atmosphere,

Â it doesn't go down to the ground.

Â And it turns out that it has a big impact on the temperature on the ground but

Â it has no impact on the top layer, which is again, still 255 Kelvin.

Â So it's this very useful conceptual thing to keep in mind and

Â it's also algebraically, much simpler to solve.

Â Because once you have the top temperature here,

Â you can take the budget for the atmosphere here and very easily solved for

Â the temperature of the ground given this temperature of the atmosphere.

Â It turns out the temperature of the ground is equal of the temperature

Â of the atmosphere times this factor of the fourth root of 2,

Â which is about 1.189, so about 20% warmer.

Â So what's happening is that in our kitchen sink analogy

Â where the water is at its steady state level in the sink and

Â enough to drive it down the drain as fast as it's coming in from the faucet.

Â It's as though, a little piece of carrot came and

Â got stuck on the drain filter there.

Â And it doesn't plug it up completely, cuz then the sink would flood and

Â the analogy would blow up and it would be no good for anybody.

Â But it sort of partially obstructs it and

Â it means that the water has to try harder to get out.

Â And so what happens then is that the water level compensates for that eventually,

Â not instantaneously, but eventually, it'll build up to a new equilibrium

Â with a higher water level where the energy flexes are balancing again.

Â So the water didn't come from a giant bucket full,

Â the carrot didn't come with a bunch of water in it, nor

Â did the pane of glass come with a bunch of energy in it.

Â But it just traps the energy or the water in this case and

Â allows it to build up to a newer higher concentration.

Â So if we come back to our table of the Goldilocks planets here,

Â Venus, Earth and Mars.

Â