Our first climate model was too cold.
And the remedy to that, is to add the greenhouse effect.
And we're going to do that in a way that's very, very simple.
You wouldn't understand the greenhouse effect, without understanding
the simple model that I'm going to show you.
But of course, reality is much more complicated.
So after we put together this very simple model and understand it
we'll be spending a fair bit of time in the
class actually talking about how to make it more realistic
to make it more like the real world.
So the bare roof model that we looked at last time, was basically this.
It had sunlight coming in, and energy leaving, according to
epsilon sigma temperature of the ground to the 4th power.
For our new model with the greenhouse effect,
we're going to suspend a pane of glass
above the ground.
And the glass has a property, that sunlight can go
through it without any impediment, it's not absorbed by the glass.
But any infrared that comes from the ground,
is going to be absorbed by that pane of glass.
And the glass itself is going to shine in the infrared in both directions, up
and down according to its own temperature. The temperature
of the pane of glass, or as I've written, the atmosphere.
The temperature of the atmosphere here.
So we're assuming throughout all of this, that these epsilon values are
equal to one, so that means that the glass has all of
the oscillators that it needs to make a nice smooth black body
curve and create the full sigma T to the 4th, black body spectrum.
Of course, it's not a perfect black body because, if it were,
it would absorb the sunlight, too.
According to the definition, it should absorb and emit all frequencies.
So it's kind of selective infrared kind of a black body.
And we're going to solve for the equilibrium temperature such that
the energy is in steady state, just like we did before.
We're going to look for the energy in and equate that to the energy going out.
And there are multiple places where we can construct a
budget in this new model because there's actually multiple temperatures.
There's a temperature of the ground and a temperature of the atmosphere.
So, starting from the ground, since we are sort of ground based
beings after all, we can just look at the arrows going into and
coming out of the ground, and write them as inputs and outputs.
What's coming into the ground here, we've got the same sunlight as
before and we have this arrow of infrared coming down from the sky.
That's epsilon, sigma, temperature
of atmosphere to the 4th power.
That's what's coming in.
And then, what's going out of the ground
is sigma T, temperature of the ground to the 4th power there.
So, that's not really as convenient as the last equation
we looked at, because we have two unknowns in it.
We can't solve for both of them at the same time with just one equation.
If you want to solve for two unknowns, you need two constraints, two equations.
So, here's another one, this is the budget for the pane of glass.
So we have what
comes into the pane of glass, is not the solar energy at all, it is in fact the
epsilon, sigma, temperature of the ground to the 4th power.
That's this arrow coming into the glass there and then what's leaving the pane of
glass is epsilon, sigma, temperature of the pane of glass to the 4th power.
And there's this
factor of two because the light is going both upward and downward.
This two is not the most intuitive thing,
you wouldn't probably have thought of it, but
without that factor of two there, there actually
wouldn't be a greenhouse effect, as we'll see.
So this is now an equation that has two unknowns in it,
temperature of the ground, and the
temperature of the atmosphere, rather haphazardly written there.
and so we could,
in principle, take these two equations and use
algebraic substitution to solve for just one of
the temperatures and then use that, plug that back in to solve for the other one.
And you're welcome to do that if you like, to try your algebra chops.
But there's an easier way to do it, and it actually has a conceptual benefit as well.
We can draw a different budget.
it is a budget for the earth system overall.
So we can draw a line,
a boundary to space, above the atmosphere and the
energy crossing this boundary to space coming down, has to
balance the energy crossing the boundary to space going up.
In a steady state, everything has to be in balance.
And so if we write those equations,
what's coming down, is just our familiar sunshine.
L times 1 minus the albedo, over 4.
And then what's going out is epsilon,
sigma, temperature of the atmosphere to the 4th.
This equation should look familiar to you.
This is actually the same equation that we got
from the last model for the temperature of the ground.
Only now, it's that the temperature of the upper layer.
And it turns out, this is sort of a general property of these simple models
and also of the more complicated earth system that, the temperature where
the energy shines out to space is, is kind of a fulcrom point.
it's fixed by how bright the sunshine
is and how much is reflected away in the albedo.
So this is the
naked earth model and the temperature of the ground is 255 Kelvin.
Here is our greenhouse model with one pane of glass here
and the temperature of the pane of glass is 255 Kelvin again.
in the exercises, you're going to work
out the balance of this, super greenhouse model.
It has two panes of glass, so twice as much greenhouse forcing as that.
What you will find, is that the top layer is 255 Kelvin
and you'll also work out a nuclear winter scenario.
Where there's crud in the atmosphere, and so the sunlight gets
absorbed in the atmosphere, it doesn't go down to the ground.
And it turns out, that has a big impact on the temperature of the ground,
but it has no impact on the top layer, which is again still 255 Kelvin.
It's a very useful conceptual thing to keep in mind,
and it's also algaebraically much simpler to solve,
because once you have the top temperature
you can take the budget for the atmosphere
and very easily solve for the temperature
of the ground given this temperature of the atmosphere.
It turns out the temperature of the ground is equal to the temperature of the atmosphere,
times this factor of the 4th root of 2 which is about 1.189,
so about 20% warmer.
What's happening is that in our kitchen sink analogy, where the
water was at its steady state level in the sink,
and enough to drive it down the drain as fast as it's coming in from the faucet,
it's as though a little piece of carrot came and got stuck on the
drain filter there.
And it doesn't plug it up completely because then, the sink would flood
and the analogy would blow up and it would be no good for anybody.
But it sort of partially obstructs it, and it means
that the water has to try harder to get out.
And so what happens then is that,
the water level compensates for that eventually, not
instantaneously, but eventually, it'll build up to
a new equilibrium with a higher water level,
where the flows balance again.
The water didn't come along with the carrot,
nor did the pane of glass come with a bunch of energy in it.
But it just traps the energy, or the water,
and allows it to build up to a newer, higher value.
If we come back to our table of the Goldilocks planets here, Venus, Earth and
Mars, here is now the temperature of the one layer model,
here the temperature of the ground when you have one pane of glass over it, and in all
cases, is warming up, by about, 20%, which is that factor of the 4th root of 2.
So, you can see that for the
earth, this greenhouse effect is a little more powerful than the
reality, that's true also for Mars, but for Venus, we need
a much more powerful greenhouse effect to build up this 700
degrees Kelvin, which is hot enough to melt lead, on Venus.
so, it's sort of an approximation to the greenhouse
effect but we will learn about the complexities and
the things that can make it stronger or weaker as we go on