Earth Calculus

This is why I started the EarthCalc blog over the summer -- teaching is always like an oncoming train! With the other things I'm still working on, the weeks have been crazy!! Hopefully things will calm down as the semester progresses.

Well, my running in the mountains went pretty well. I'm a slow runner and the 11,000 foot altitude made me slower. This issue of getting enough oxygen is pretty important.

My previous worksheets on altitude and pressure talked about linear functions, power functions versus exponentials, and deriving the equations themselves, which was pretty sophisticated. This one has two pages (if printed duplex) asking basic questions about one-to-one functions and inverse functions. Interestingly, while atmospheric pressure decreases with altitude (monotonically if not constantly), temperature decreases and then increases due to the sun's warmth. Funky stuff happens at the very outer edges of the atmosphere, which we don't discuss. (But check out the leaky atmosphere worksheet if you're interested!)

I'll put up answers to this soon but wanted to get the worksheet out first. Here it is:

Inverses: Altitude

This week's post is

•  the first this semester (fall semester! I am teaching again! I wasn't quite sure if I would or not!)

• curiously appropriate, since I am up in the mountains this weekend.

I'm at Brian Head Resort briefly, running down a mountain. This running race starts at around 10,000 feet of elevation and I'm not used to it. There's a short initial uphill run, and then a chairlift ride, and then a lot of downhill. I am thinking keenly about the amount of oxygen available to my brain and muscles.

(I'm hearing from my running-mates that you can't drive a rental car up Mauna Kea -- they all die because the air is thin enough that combustion is affected. You've got to get a good SUV. So throw this out to your students as a real-life application!)

Regular readers might remember the first set of worksheets I did on altitude. As I come back to class this September I'm reminded that there's always room for working through the most basic concepts in a clear and straightforward way. The worksheet I'll post in a day or two (next internet access!) looks at the concepts around inverse functions: the horizontal and vertical line tests. It uses some data taken from empirical measurements of temperature at different altitudes. Remember that we're hot down here, the temperature drops as we climb mountains and fly up in planes, but then if we're on the sunny side of the planet the temperature rises again in the upper atmosphere. There are some wiggles in the middle, too. (Check out the graph at this site.) Clearly, me telling you what temperature I am does not tell you uniquely what altitude I'm at! On the other hand, pressure is monotonically decreasing as altitude increases, and so me telling you an atmospheric pressure would allow you to estimate my altitude.

More later: time to adjust to 10,000 feet!

Check out this beautiful visualization of seasonal land ice and vegetation variation at NPR or at the original source:

Breathing earth

I was traveling last week for a delightful conference on quantum cohomology, affine Schubert calculus, and the Peterson isomorphism. Crazy stuff. I had somehow expected to post during that -- but I learned something, which is that posting does not happen during intense conferences. I'll deal with that differently next time!

Many weeks ago in a discussion on LinkedIn someone suggested I write a post on sea ice, and I've been reading about sea ice ever since. It is complicated! Sea ice is the ice in the polar regions that floats on the sea, of course. It plays a huge role in climate because of its reflectivity, and we watch the growth and shrinkage of sea ice to monitor the state of the Arctic and Antarctic. But sea ice isn't just ice like an ice cube. Because the sea is salty, and salty water and fresh water freeze at different temperatures, there are some complicated phenomena that go on.

In the sea ice there are polynyas, areas of open water on top of the ice, and brine channels filled with bacteria and other little creatures. The ice itself ranges from new ice, starting with frazil ice through nilas ice through the stuff we think of when we think "ice." Even the names are exotic. Check out this picture below from the Advanced Land Imager (ALI) on NASA’s Earth Observing-1 (EO-1) satellite:

Zoom in to see labels for frazil and nilas.

One of the sites I'm using for a resource for the coming worksheet has a lot more cool pictures, so check it out.

With all this complexity in biology, climate, physics, and thermodynamics, there's a bit of work involved in simplifying the models for a calculus worksheet. The worksheet will get done by Thursday and will focus on Stefan's model (from 1890!) for the thickness of sea ice. It will use a bit of integration and some differential equations.

Assumptions need to be made to produce a model this simple: I'm going to assume no snow on top of my ice, no brine channels or microorganisms messing up my idealized slab, a linear temperature gradient inside the ice. Even with these assumptions we can make rough predictions about how sea ice thickness evolves.

But, so that I don't leave you without any classroom resource today, I leave you with a link to a nice precalculus activity on modeling ice thickness. I just discovered Plus Magazine, the source, today. It's a nice place to look for short articles on beautiful mathematics!