Modeling Earth’s Energy Balance (Zero-D) (Equilibrium)

For conservation of energy, the short-wave solar energy absorbed by the Earth equals the long-wave outgoing radiation.

Energy and matter can’t just disappear. Energy can change from one form to another. As a thrown ball moves upwards, its kinetic energy of motion is converted to potential energy due to gravity. So we can better understand systems by studying how energy (and matter) are conserved.

Energy Balance for the Earth

Let’s start by considering the Earth as a simple system, a sphere that takes energy in from the Sun and radiates energy off into space.

Incoming Energy

At the Earth’s distance from the Sun, the incoming radiation, called insolation, is 1367 W/m². The total energy (wattage) that hits the Earth (E_in) is the insolation (I) times the area the solar radiation hits, which is the area a cross section of the Earth (A_cx).

$E_{in} = I \times A_{cx}$

Given the Earth’s radius (r_E) and the area of a circle, this becomes:

$E_{in} = I \times \pi (r_E)^2$

Outgoing Energy

The energy radiated from the Earth is can be calculated if we assume that the Earth is a perfect black body–a perfect absorber and radiatior of Energy (we’ve already been making this assumption with the incoming energy calculation). In this case the energy radiated from the planet (E_out) is proportional to the fourth power of the temperature (T) and the surface area that is radiated, which in this case is the total surface area of the Earth (A_surface):

$E_{out} = \sigma T^4 A_{surface}$

The proportionality constant (σ) is: σ = 5.67 x 10^-8 W m^-2 K^-4

Note that since σ has units of Kelvin then your temperature needs to be in Kelvin as well.

Putting in the area of a sphere we get:

$E_{out} = \sigma T^4 4 \pi r_{E}^2$

Balancing Energy

Now, if the energy in balances with the energy out we are at equilibrium. So we put the equations together:

$E_{in} = E_{out}$

$I \times \pi r_{E}^2 = \sigma T^4 4 \pi r_{E}^2$

cancelling terms on both sides of the equation gives:

$I = 4 \sigma T^4$

and solving for the temperature produces:

$T = \sqrt{\frac{I}{4 \sigma}}$

Plugging in the numbers gives an equilibrium temperature for the Earth as:

T = 278.6 K

Since the freezing point of water is 273K, this temperature is a bit cold (and we haven’t even considered the fact that the Earth reflects about 30% of the incoming solar radiation back into space). But that’s the topic of another post.

Sling: A VPython Model Demonstrating Centripetal Force and Conservation of Angular Momentum

sling-anim-500 — Animation captured from the sling.py Vpython model. The yellow arrow shows the centripetal force. The white arrow shows the velocity.

Sitting in a car that’s going around a sharp bend, its easy to feel like there’s a force pushing you against the side of the car. It’s called the centrifugal force, and while it’s real to you as you rotate with the car, if you look at things from the outside (from a frame of reference that’s not rotating) there’s really no force pushing you outward. The only force is the one keeping you in the car; the force of the side of the car on you. This is the centripetal force. Given all the potential for confusion, I created this little VPython model that mimics a sling.

Centripetal Force

In the model, you launch a ball and it goes off in a straight line. That’s inertia. An object will move in a straight line unless there’s some other force acting on it. When the ball hits the string, it catches and the string starts to pull on the ball, taking it away from its straight line trajectory. The force that pulls the ball away from its original straight path is the centripetal force.

centrifA — Image from Stern (2004): (23a) Frames of Reference: The Centrifugal Force

Conservation of Angular Momentum

The ball rotating on the sling has an angular momentum (L) that’s equal to the velocity (v) times its mass (m) times its radius (r) away from the center.

L = mvr , angular momentum

Now, angular momentum is conserved, which means that if you shorten the string, reducing the radius, something else must increase to compensate. Since the mass can’t change, the velocity has to, and the ball speeds up.

momentum-diag — The ball on the string with the shorter radius has the higher velocity (moves faster). It also has a higher centripetal force. The ball for shortening the radius is not shown in this figure.

I’ve put in a little ball at the end of the string that you can pull on to shorten the radius.

Tangential Velocity

Once the ball is attached to the string, the centripetal force will keep it moving in a circle. If you release the ball then it will fly off in a straight line in whatever direction it was going when you released it. With no forces acting on the ball, inertia says the ball will move in a straight line.

The ball, when released from the string, flies off in a tangent.

To better illustrate the ball’s motion off a tangent, I put in a target to aim for. It’s off the screen for the normal model view, but if you rotate the scene to look due north you’ll see it.

POV-Ray: 3d rendering

Giles Tran’s amazing rendering of glasses on the counter inspired me to check through my own POV-Ray generated library. Nothing nearly as good, but some of it is still might be useful.

Axial_tilt — Demonstrating the axial tilt of the Earth, this image shows the Earth at the northern hemisphere's summer solstice.

120px-ElipticN_000_2xAplaneClose — Rotating Earth at the northern hemisphere's winter solstice.

You build 3d models in POV-Ray and then export 2d images from whichever point of view you want, so once you have the model set up you can easily change the perspective or even move objects to create animations.

POV-Ray does not have any useful sort of user interface; you’re usually creating your models with computer code. It can therefore be challenging to use, and, as with any 3d programming language, a bit of geometry, trigonometry and algebra are needed.

However, the final results can be impressive. I’m continually amazed each year by the quality of the work added to their Hall of Fame.

For much easier, quicker and not so sophisticated 3d results, I use VPython, which is also a great way to learn programming that outputs 3d images.

Urban planning with SimCity

The SimCity game is a wonderful model for urban planning. My class is using it to try to tie together the lessons on the Needs of People and the Themes of Geography.

I gave the small groups the game, two hours, and required them to take notes on why they made the choices they made.

SimCity_4-_Rush_Hour_region_view — SimCity regional view.

What we did

The game starts at the Region view, where you choose the location of the city. I was enthused to see the groups almost instinctively go for a location with good access to water. Of course almost all the places you can found a city are on a river or ocean, but more than one student specifically mentioned the water access as a reason for their choice.

To have them better think about the region, I also asked the students to think about, and report, on where in the world they thought their city might be, based on the topography and the vegetation. Most proposed the eastern U.S. seaboard.

After choosing a location the students could “terraform” it by raising mountains, making valleys, sculpting beaches and more. Some groups needed to be chivvied to move on, after all, they only had one two hour session to complete the assignment.

Then they got into the heart of the game, Mayor Mode (the terraforming session is called “God Mode”). The urban planning model is based on the land-use zoning strategy used by many, but by no means not all, U.S. cities. You have to mark cells on the city’s grid for residential, commercial or industrial/agricultural use. Then, if you’ve provided utilities and a transportation system “developers” will autonomously start to build houses, businesses and industry in these zones.

Da Hood-4 Mar., 041286566223 — The great city of Da Hood. Note the different areas for urban, commercial and industrial development, and the seaport on the river.

Playing on “Easy”, the mayoral advisers would regularly pop up to suggest new amenities, like schools, police stations and parks that would attract more people to the city.

And students had to make choices. One of the first, for example, was about what type of power to provide their city. Coal plants are cheap but dirty, while windmills produce a lot less power so you have to build a lot of them.

A Little Discussion

The game worked remarkably well as part of the curriculum. SimCity is a potentially addictive game, the plea, “I really need to stop,” was heard repeatedly as I was trying to get the last group to come to our discussion. Yet, two hours was enough for students to get the gist of the game and think about its implications for geography. The final cities were not perfect (at least one was designed to be dysfunctional) and most of them were running a serious deficit, but when it came time to present, students were able to flesh out our information on the lessons quite nicely.

The game is also easy enough. The game’s internal model is quite sophisticated, but there’s enough in-game advice, that it took just some initial guidance about the basic premise of zoning, for students unfamiliar with the game to play it effectively. Some students were better prepared at the start than others. Some had played similar games in the past and one student had even read the instruction booklet that came with the game CD, but they were all able to get cities up and running in the allotted time.

Technical Difficulties

We’re a Mac school, but SimCity does not have a version that works with modern macs, so I had to use my old laptop that has Windows. That computer is a Mac that it uses Boot Camp to boot to Windows, and, perhaps for this reason, the first group that tried to use it had it crash on them a few times at the beginning of their game. They gave up and created their city in our sandbox, which turned out great in the end because it gave them more flexibility in the structures they could create and some interesting differences in perspectives from the game based presentations. I’ll post more about that later.

In Conclusion

I like the game because it lets the students provide the infrastructure while the game engine/model tests the infrastructure to see it if works and “predicts” development and population. The Needs of People and Themes of Geography contexts were useful ways of getting students into the game but struggling to get the city to work helped fill in a lot of things that students had not thought of previously.

One of those things was people’s need for safety. In our post-game discussion, safety from crime and from nature came up as additional needs of people we had not discussed. Successful cities in the game need police stations, and students had apparently been thinking hard about the array of natural disasters they could rain down on their cities when the assignment was over.

Da Hood-18 May., 041286566502 — The Taj Mahal, soccer fields and a skate-park (of which some of us were inordinately proud) met the needs of citizens for recreation and understanding.

Finally, students presented their cities while Ms. Ann DeVore from the Deargorn Heights Montessori Center was observing the classroom. Ann is an enthusiastic user of SimCity. Her middle school uses it the initial part of the Future City competition, which is something I’d very much like to get my group involved in as soon as I can wrangle some technical advisers.

Seeing temperature, kinetic energy and color

We read that temperature is the average kinetic energy of a substance but you can (especially if you’re a visual learner) nicely internalize this from simple videos or animations. UCAR has a little animation with their definition of temperature. I however, adapted an interactive, 3d animation that I think does a nice job, and also introduces a couple of other interesting concepts too.

I’ve also used this model, at different times, to show:

The relationship between temperature and color emitted by objects. The main way we know the temperature of stars is because blue stars are hotter than red stars. Blue light has a shorter wavelength than red light, and things that are at higher temperatures emit shorter wavelengths.
Absolute zero (0 Kelvin) – where (almost) all motion stops and the objects stop emitting light.
Pressure in a gas – you really get a feel for the force exerted by the particles on the side of the box (although it might be even more interesting once I figure out how to add sound).

It is an interactive model, but it’s pretty simple because the only control is a slider that lets you set the temperature.

Finally, in the age of 3d movies, like Avatar, the models can be easily shown in 3d if you have the glasses (redcyan).

The model is easy to install and run on Windows, but you have to install the programming language VPython separately on a Mac (but that isn’t very hard). I have this, and a bunch of other models, at http://earthsciweb.org/GeoMod/.