Monthly Archives: July 2013

How to survive a climate catastrophe

How fun to see something on the Scientific American Blogs that is so appropriate to what we've been writing about! The writer, Zev Brook, is a high school student entering 12th grade:

How to survive a climate catastrophe

Alright, he doesn't really tell us how to survive a climate catastrophe, but the first half is a nice story of the science around the Paleocene-Eocene Thermal Maximum, something mentioned briefly by the speakers during the conceptual climate models seminar as a topic they would have loved to discuss. Something Zev Brook mentions was brought up by the speakers in the final Q&A session, too: the earth is going to survive almost anything we do just fine. It's ourselves we have to worry about.

Snowball earth... last talk at climate model summer course

These are notes from the last talk of the MAA North Central Section-sponsored summer seminar on conceptual climate models. This talk by Anna Barry tied together all the things we'd learned about over the past two days in discussing the snowball earth hypothesis, which tries to explain some mysterious pieces of paleoclimate evidence, and whether or not there is a mathematical basis for the idea.

So, let's get started!

What could initiate a snowball earth state?

Ice-albedo feedback, which we discussed earlier (more ice -> higher albedo (more reflectivity) -> less energy in, as more solar energy is reflected -> colder -> more ice).
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Understanding climates past: more from the summer seminar

More notes from the MAA-sponsored North Central Section summer seminar on conceptual climate models. This is from Richard McGehee's talk on understanding the climate of the past and the Milankovitch cycles. These notes give some overview, but the graphs are really important to understanding these ideas and I will work on finding some to include.

Some thought-provoking questions: If we can’t even predict the weather, how can we predict the future? If we don’t know about the climates of the past, how can we expect to predict the future? The question is somewhat controversial: some climate modelers feel we only need to understand today and then we can play it all forward using big general climate models.

How do we know the climates of the past?

Lake Vostok, Antarctica. 2.2 miles of ice on top of a tiny little pool of water down near the earth. Scientists have taken core samples from here and “gone back in time.” “Isotopes in the ice are proxies for past atmospheric temperatures above the Antarctic”: Continue reading

Greenhouse gases: more blogging from MAA-NCS climate course

Jim Walsh from Oberlin opened today by talking about greenhouse gases and energy balance equations. His slides are online -- check them out for all the great pictures I have not included!

First big conceptual point: global climate is determined by the energy in minus the energy out. Since energy in is basically the insolation ( Q -- incoming solar radiation) that is not reflected (multiply by 1-albedo) and energy out is OLR (outgoing longwave radiation) these are the three factors to look at -- change in insolation, albedo, or OLR. If these are changed by our human activities (or anything else!) climate will change.

Here Jim talked about the Earth Radiation Budget Experiment briefly.

Energy balance and greenhouse gases

Radiation is characterized by its direction of propagation and frequency \nu . We need to know about electromagnetic spectrum, and for climatology (look at Pierrehumbert's book, p137) we need infrared through ultraviolet.
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Energy balance models at the MAA NCS course on climate modeling

Some quick notes from Esther Widiasih's talk at the MAA North Central Section summer seminar on climate modeling -- thanks again to the MAA and MCRN for sponsoring the workshop!

Start with the Budyko's energy balance model (EBM) -- a linearized version:

 R \frac{dT}{dt} = Q (1-\alpha) - (A+BT)

 with equilibrium solution T_{eq} = \frac{Q(1-\alpha)-A}{B}.

This equilibrium is stable with eigenvalue -B (recall B >0 ).

What if the earth’s albedo was not 0.3 ? Remember, albedo of ice is 0.62 , so changing ratios of ice to land to water change overall albedo.

Next step: zonal energy balance models.

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Sage worksheet on energy balance model

Attached is a Sage worksheet on the simplest global energy balance model. An energy balance model looks at energy in and energy out:

change in temperature = energy in - energy out

Pretty straightforward, eh? Let Q be the incoming solar radiation (insolation) and \alpha be albedo (percentage of sunlight the earth reflects) -- our "energy in" will be Q(1-\alpha) . Then we can use Boltzmann's black-body radiation to get "energy out" -- it's  \sigma T^4 , where T is our global average surface temperature in kelvins and \sigma is Boltzmann's constant. So we get the equation

\frac{dT}{dt} = Q(1-\alpha)-\sigma T^4

It is not too hard to use calculus to find a linearization near the equilibrium point of this equation and then do some analysis.

The attached worksheet can be loaded into if you want to work through it without installing Sage:

MAA-NCS Climate Modeling -- Boltzmann

I'm trying to figure out how to put up a full interactive worksheet; it's not so hard to put up cells, but I'm not sure about a whole Sage file... let me know if you know!

Live-ish blogging the MAA NCS Climate Modeling course

The MAA North Central Section is having a summer short course on climate modeling. This morning we've started out with an overview of climate and climate modeling by Samantha Oestreicher. We'll be alternating between lectures and hands-on computer modeling.

I'll be trying to live-blog it, more or less.

Here goes! Some notes from Samantha's talk.

What is climate? Climate versus weather:  "Do I need to own an umbrella?" versus "Do I need an umbrella today?"

How do we observe climate? Data comes from many sources:
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Derivation of equation for atmospheric pressure

Flashbacks to the past: one of the first worksheets I published in this project was on using linear approximation to estimate the atmospheric pressure at various altitudes, and a later one was about a power function for atmospheric pressure. The derivation of the formula for atmospheric pressure is actually pretty straightforward. I'll assume that your students have not yet encountered integrals per se, but this worksheet pushes them to use their knowledge of differentiation to deduce an antiderivative.

This is a worksheet that puts together a few disparate concepts:

  • dimensional analysis, using units to understand equations
  • antiderivatives,
  • and baby differential equations.

It's certainly an activity for the end of the section on differentiation. The very last question asks students to think about a more accurate equation, and I wouldn't expect most students to be able to solve it alone -- but sometimes a good challenge is important as it points to concepts you'll be dealing with later on. Knowing how to integrate would really help in solving that last problem 🙂

Derivation: Atmospheric Pressure

As mentioned in the earlier posts, a great resource on atmospheric pressure and rocketry and all sorts of fun things is found at Portland State Aerospace Society's rocketry pages.

Related Rates: Compost oh compost

I'm on a nitrogen kick, I guess. While researching the previous post on beets, which ought to have a follow-up when we get to integration (did you do the worksheet?!), I learned a bit about carbon-nitrogen ratios in compost. The ratio of carbon to nitrogen is important because compost gets hottest when this ratio is around 30:1. Empirical evidence supporting this is not too hard to find, but a mathematical model is hard to find! This is why you need to mix "browns" and "greens" in your home compost.

Again, nitrogen is a very important nutrient for vegetable and grain growth -- if you use up the nitrogen from your soil you'll have small flowers and no tomatoes. Nitrogen conservation in your compost is thus pretty important. But again, I had a hard time finding math equations for this -- unless you count all the cool papers that solved twelve nonlinear systems simultaneously.

Finally, after days of trolling through the library files and becoming much more aware of what a HUGE BUSINESS waste processing is, I found a paper with some polynomials. It's also got some great 3D graphs and some visualizations, in Figure 4, of how these functions depend on their independent variables. Might be food for another post. The authors of "Optimizing composting parameters for nitrogen conservation in composting" took an approach similar to the beet-research people: they did a bunch of experiments, measuring values for aeration of the compost heap, moisture content, particle size, and time after start, and ran a big backward computation to come up with polynomials in those variables (A, M, P, and t) that predicted the carbon/nitrogen ratio (among other things) pretty well. That is the polynomial that the worksheet below focuses on.

This worksheet covers

  • related rates, applying the product and chain rules a few times
  • physical reasoning: I ask students to examine the assumptions of the model, which I then violate for mathematical purposes!
  • and writing English sentences explaining a math result.

Teaching tips: students often freak out about all the symbols in here. Reassure them that many are constants in the problems they're asked to work out. Remind them that the derivative of a constant is zero, even if the constant is one they don't know!

I would like to have had some other "real-life scenarios" or more interpretation, so I'll think about what sorts of related-rates problems could be added to this.

Related Rates: Nitrogen

If you've got refinements or modifications, let me know!

Give me some optimal sugar... calculus-style

I was on a math trip this week, so have been a bit delayed in posting. After four days of intensive pure math thought, I've returned to my little farm in the city, the minuscule plot that is my back yard. Today was spent doing math and picking cherries. The beans and peas are doing well, too; we've got peas planted where we used to grow tomatoes in an effort to increase the nitrogen content of the soil without applying fertilizer.

I have the luxury of not depending on my little garden for my primary food source. Instead, I buy food from farmers either at the farmer's market or at the grocery store. It is nice to live in the city and be able to take the bus to the opera, but it means I depend on others for agriculture. They use fertilizer and irrigate their land because farmers must do everything they can to control growing conditions for their crops.

Today is about sugar beet production. (I also looked into optimization of conditions for composting, but there are no equations I can find!) Sugar beets are a major crop across the US, particularly in North Dakota, Minnesota, and Idaho. We love sugar and want it in many foods (until we find it's killing us) and of course farmers want to optimize their yields. Sugar beets are interesting because simply adding more nutrients to the soil can be counterproductive: you don't want the biggest sugar beets, you want the sweetest ones! Too little nitrogen means yellow leaves and poor growth. Too much nitrogen means impurities in your beets and reduced sucrose, or at worst killing your seedlings (source). It's the Goldilocks question.

An older report on how nitrogen levels affect recoverable sugar yields has some very nice equations. G.L. Malzer and Greg Buzicky looked at many variables and came up with several equations that predicted recoverable sugar yield pretty well, all in terms of the soil's nitrate-nitrogen content. And they're quadratic! This is a nice way to do a pretty easy optimization exercise with applications to something... sweet!

The first page is all about finding the optimum recoverable sugar yields given different levels of nitrate-nitrogen in the soil. The second page mixes in some experimentation and treats a two-variable function, foreshadowing multivariable calculus techniques. Including discussion of multivariable functions in a first-semester calculus course is a really cool idea that deserves more attention -- it does not disturb students, but only people who have a set idea of what one "should" learn in first-semester calculus. The third page asks students to use the first and second derivative tests to prove the results they've already produced, and asks them to think about the applicability of the Extreme Value Theorem. As in many situation, physical constraints could lead to a closed-interval phrasing of the problem, although it's not necessary mathematically. Provoke an argument!

Optimization: Sugar Beets

Agriculture is often ignored in calculus and STEM classes, as it's not so sexy these days. Universities like the University of Minnesota and Cornell have big ag programs, though, and they're hugely important. You sure can't be a vegan or vegetarian in the north without the products of modern agriculture!

I've learned a lot from my ag students and they need to deal with optimization often: they need to optimize nutrient composition in animal feed, optimize nutrient composition of fertilizers for soybean growth, optimize temperature for dry-matter intake of chickens, and of course look at the economics of all the above. Their decisions impact the diets, waterways, air quality, climate, and fuel prices of city-dwellers. Don't forget the ag, even if you live in LA.