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Our first climate model was too cold.

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And the remedy to that, is to add the
greenhouse effect.

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And we're going to do that in a way that's
very, very simple.

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You wouldn't understand the greenhouse
effect, without understanding

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

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But of course, reality is much more
complicated.

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

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we'll be spending a fair bit of time
in the

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class actually talking about how to make
it more realistic

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to make it more like the real
world.

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So the bare roof model that we looked at
last time, was basically this.

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

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epsilon sigma temperature of the ground to
the 4th power.

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For our new model with the greenhouse
effect,

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we're going to suspend a pane of glass

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above the ground.

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And the glass has a property, that
sunlight can go

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

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But any infrared that comes from the
ground,

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is going to be absorbed by that pane of
glass.

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And the glass itself is going to shine in
the infrared in both directions, up

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and down according to its own temperature.
The temperature

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of the pane of glass, or as I've written,
the atmosphere.

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The temperature of the atmosphere here.

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

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

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

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curve and create the full sigma T to the
4th, black body spectrum.

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Of course, it's not a perfect black body
because, if it were,

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it would absorb the sunlight, too.

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

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

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And we're going to solve for the
equilibrium temperature such that 

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

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We're going to look for the energy in and
equate that to the energy going out.

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And there are multiple places where we can
construct a

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budget in this new model because there's
actually multiple temperatures.

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

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

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

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coming out of the ground, and write them as inputs and outputs.

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What's coming into the ground here,
we've got the same sunlight as

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

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That's epsilon, sigma,
temperature

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of atmosphere to the 4th power.

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That's what's coming in.

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And then, what's going out of the ground

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is sigma T, temperature of the ground to the
4th power there.

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So, that's not really as convenient as the
last equation

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we looked at, because we have two unknowns
in it.

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

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If you want to solve for two unknowns, you
need two constraints, two equations.

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So, here's another one, this is the budget
for the pane of glass.

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So we have what

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comes into the pane of glass, is not the
solar energy at all, it is in fact the

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

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That's this arrow coming into the glass
there and then what's leaving the pane of

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glass is epsilon, sigma, temperature of
the pane of glass to the 4th power.

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And there's this

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factor of two because the light is going
both upward and downward.

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This two is not the most intuitive thing,

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you wouldn't probably have thought of it,
but

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without that factor of two there, there
actually

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wouldn't be a greenhouse effect, as we'll
see.

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

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temperature of the ground, and the

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 temperature of the atmosphere, rather haphazardly
written there.

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and so we could,

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in principle, take these two equations and
use

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algebraic substitution to solve for just
one of

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the temperatures and then use that, plug
that back in to solve for the other one.

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

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But there's an easier way to do it, and it
actually has a conceptual benefit as well.

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We can draw a different budget.

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it is a budget for the earth system overall.

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So we can draw a line, 

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a boundary to space, above the atmosphere
and the

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energy crossing this boundary to space coming
down, has to

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balance the energy crossing the boundary
to space going up.

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In a steady state, everything has to be in
balance.

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And so if we write those equations,

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what's coming down, is just our familiar
sunshine.

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L times 1 minus the albedo, over 4.

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And then what's going out is epsilon,

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sigma, temperature of the atmosphere to
the 4th.

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This equation should look familiar to
you.

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This is actually the same equation that we
got

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from the last model for the temperature of
the ground.

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

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

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and also of the more complicated earth
system that, the temperature where

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the energy shines out to space is, is kind
of a fulcrom point.

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it's fixed by how bright
the sunshine

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is and how much is reflected away in the
albedo.

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So this is the

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naked earth model and the temperature of
the ground is 255 Kelvin.

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Here is our greenhouse model with one pane
of glass here

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

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in the exercises, you're going to work

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out the balance of this, super greenhouse
model.

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

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What you will find, is that the top layer is
255 Kelvin

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and you'll also work out a nuclear
winter scenario.

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Where there's crud in the atmosphere, and
so the sunlight gets

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absorbed in the atmosphere, it doesn't go
down to the ground.

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And it turns out, that has a big impact on
the temperature of the ground,

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

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It's a very useful conceptual thing 
to keep in mind,

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and it's also algaebraically much simpler to solve,

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because once you have the top
temperature

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you can take the budget for the
atmosphere

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and very easily solve for the
temperature

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of the ground given this temperature of
the atmosphere.

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It turns out the temperature of the ground is
equal to the temperature of the atmosphere,

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times this factor of the 4th root of 2
which is about 1.189,

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so about 20% warmer.

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What's happening is that in our kitchen
sink analogy, where the

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water was at its steady state
level in the sink,

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and enough to drive it down the drain as
fast as it's coming in from the faucet,

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it's as though a little piece of carrot
came and got stuck on the

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drain filter there.

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And it doesn't plug it up completely
because then, the sink would flood

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

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But it sort of partially obstructs it, and
it means

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that the water has to try harder to get
out.

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And so what happens then is that,

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the water level compensates for that
eventually, not

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instantaneously, but eventually, it'll
build up to

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a new equilibrium with a higher water
level,

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where the flows balance again. 

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The water didn't come along with the carrot,

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nor did the pane of glass come with a bunch of energy in
it.

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But it just traps the energy, or the water,

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and allows it to build up to a
newer, higher value.

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If we come back to our table of the
Goldilocks planets here, Venus, Earth and

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Mars, here is now the temperature of the
one layer model,

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here the temperature of the ground when you
have one pane of glass over it, and in all

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cases, is warming up, by about, 20%, which
is that factor of the 4th root of 2.

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So, you can see that for the

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earth, this greenhouse effect is a little
more powerful than the

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reality, that's true also for Mars, but
for Venus, we need

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a much more powerful greenhouse effect to
build up this 700

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degrees Kelvin, which is hot enough to
melt lead, on Venus.

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so, it's sort of an approximation to the
greenhouse

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effect but we will learn about the
complexities and

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the things that can make it stronger or
weaker as we go on