Our leaky atmosphere

Backing away from the sensational headlines of last week, here's something a bit more relaxing. Our atmosphere is leaking away into space and maybe someday eventually we won't have one anymore. How's that?

Alright. I exaggerate a bit here -- the force of gravity is enough to keep nitrogen and oxygen around, as well as lovely greenhouse gases like carbon dioxide and methane. These are heavy molecules! Hydrogen, on the other hand, is really light. It reaches escape velocity fairly easily, especially at the outer reaches of the earth's atmosphere.

For a more nuanced look at atmospheric escape, check out the Scientific American article. I'm linking to an author's website because I can't get this article at the Scientific American site even via Cornell's library.

Today's worksheet is another longish one. I might consider splitting it up. The first two pages guide the student through a derivation of escape velocity using the chain rule, integration, and limits. It's one of those activities that dredges up everything a student had to learn all semester, yet in the end none of the integration or limits are that hard. The last page (the third) asks some very concrete questions about the escape velocity for hydrogen molecules, relating this velocity to temperature. "Is it hot enough for hydrogen to escape?" is the main question.

I have to confess that I spent all day traveling to a math conference and I'm a bit underslept. As I went through my calculations I realized that hydrogen could escape from anywhere in the earth's atmosphere, and this really alarmed me! We are all going to perish! until I realized that indeed that's the case, and that's why we basically have no hydrogen in our atmosphere. It's ok. Can't breathe that stuff anyway.

Let me know if you use this worksheet and whether you think it would be better broken up into parts. I like the surprising results, and I like the fact that it brings together the chain rule, integration, and limits. It would be fun to extend it and talk about the moon's lack of atmosphere and Venus's crushing anvil of an atmosphere. We sometimes take for granted our Goldilocks world...

Calc Roundup: Atmospheric Escape

Drinking drugs in Denver

How's that for a sensational headline?

This post is a bit late because of units. I got some wonderful data on personal care products and pharmaceuticals in wastewater from a reader: several pages of information on what's going into and coming out of the wastewater treatment plants in Silverthorne, Dillon, and Frisco, Colorado. These three wastewater treatment plants treat water from homes all around the Dillon Reservoir and then put treated water out into the reservoir itself, which eventually provides drinking water to Denver. They're doing the same great work wastewater treatment plants around the US and the world are doing: taking our extra coffee, our dishwater, and our waste and putting out pretty clean water!

The study I'm looking at used six trips over three years to look at influent (incoming water) and effluent (outgoing water) at these wastewater treatment sites. The authors used grab samples -- quick samples taken at one time, rather than samples taken at specific repeated times or water sampled throughout the day -- to get a cost effective test of water quality. Now, water spends some time in the wastewater treatment plant (this is called resonance time or residence time) and so a grab sample will not reflect what happens to a particular batch of water that goes in -- if it takes three hours, for instance, for water to get processed, but you take samples of influent and effluent ten minutes apart, you're measuring different parts of the stream. That will explain a bit of the weirdness we encounter on page three of the worksheet. Naproxen and cocaine are very effectively removed from water by current treatments, but hydrocodone is not and so it looks like more is coming out of the treatment plant than is going in!

Back to units, though: I wanted to do a worksheet using Riemann sums to estimate quantities of pharmaceuticals in the water supply, but all my data is in nanograms per liter and I wanted to look at it against time rather than volume. So I had to figure out how to get nanograms or grams per day, and that involved finding out how much water each wastewater treatment plant processed. Not totally easy. But I did manage it roughly. I found that in 2002-2004 the Snake River Wastewater Treatment plant was processing about 27% of its official capacity of 2.6 million gallons per day and used that figure. The Joint Sewer Authority (JSA) plant in Silverthorne says in its financial statement that it processes four million gallons/day. (I'd like to check those numbers, and have an email out to the Snake River folks.) Turn those into liters, and we can get nanograms per day. But those numbers are ugly decimals, so let's talk about grams per day!

I made two Google spreadsheets, one with data for students to use and one with the answers (for you and me, on the solutions page). This activity is best as a computer activity, I think, although if students have graph paper available I did provide the data on the worksheet pdf and it would not be toooo onerous to do the Riemann sums by hand. Up to you to manage the discussion about drugs.

 Numerical Integration: Naproxen, Hydrocodone, Cocaine

Student version of spreadsheet

I will change the numbers if the Snake River folks respond with more up-to-date information about how much water they are treating per day.

Anyhow, this worksheet deals with using Riemann sums to roughly estimate a quantity (net amount) given a rate. If you choose to make it a computer activity, it's pretty straightforward to answer all the questions in a concentrated class group-work session. If you decide to do this only on paper, you might choose not to deal with all three chemicals -- it might take more time than is appropriate for an in-class activity. Each chemical is on a different page so mix and match!

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 ( -- 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 . 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:

 with equilibrium solution

This equilibrium is stable with eigenvalue (recall ).

What if the earth’s albedo was not ? Remember, albedo of ice is , 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 be the incoming solar radiation (insolation) and be albedo (percentage of sunlight the earth reflects) -- our "energy in" will be . Then we can use Boltzmann's black-body radiation to get "energy out" -- it's , where is our global average surface temperature in kelvins and is Boltzmann's constant. So we get the equation

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 sagenb.org 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.