Solèr’s Theorem

Here’s another post on the foundations of quantum theory:

• Solèr’s Theorem.

It’s about an amazing result, due to Maria Pia Solèr, which singles out real, complex and quaternionic Hilbert spaces as special. If you want to talk about it, please join the conversation over on the n-Category Café.

All these recent highly mathematical blog posts are a kind of spinoff of a paper I’m writing on quantum theory and division algebras. That paper is almost done. Then our normal programming will continue: I’ll keep going through Pacala and Socolow’s “stabilization wedges”, and also do a This Week’s Finds where I interview Tim Palmer.

State-Observable Duality

It’s confusing having two blogs if you only have one life. I post about my work at the Centre for Quantum Technology here. I post about abstract algebra at the n-Category Café. But what do I do when my work at the Centre for Quantum Technology starts using a lot of abstract algebra?

I guess this time I’ll do posts over there, but link to them here:

• State-Observable Duality (Part 1).

• State-Observable Duality (Part 2).

• State-Observable Duality (Part 3).

This is a 3-part series on the foundations of quantum theory, leading up to a discussion of a concept I call ‘state-observable duality’. The first part talks about normed division algebras. The second talks about the Jordan-von Neumann-Wigner paper on Jordan algebras in quantum theory. The third talks about state-observable duality and the Koecher-Vinberg theorem.

I think I’ll take comments over there, so our discussion of environmental issues here doesn’t get interrupted!

Stabilization Wedges (Part 2)

Okay. We’re going through this paper, which you can read yourself:

• Stephen Pacala and Robert Socolow, Stabilization wedges: solving the climate problem for the next 50 years using current technologies, Science 305 (2004), 968-972.

The paper lists 15 ‘wedges’, each of which could ramp up to reducing carbon emissions by 1 gigaton/year by 2054. We’re going through all these wedges and discussing them. And the Azimuth Project is lucky to have a new member on board — Frederik De Roo — who is summarizing our discussion here:

• Azimuth Project, Stabilization wedges.

So, let’s get going!

Last time we covered four wedges related to energy conservation and increased efficiency. Wedge 5 is in a category of its own:

5. Shifting from coal to natural gas. Natural gas puts out half as much CO₂ as coal does when you burn them to make a given amount of electricity. After all, it’s mainly methane, which is made from hydrogen as well as carbon. Suppose by 2054 we have coal power plants working at 90% of capacity with an efficiency of 50%. 700 gigawatts worth of coal plants like this emit 1 gigaton of carbon per year. So, we can reduce carbon emissions by one ‘wedge’ if we replace 1400 gigawatts of such plants with gas-burning plants. That’s four times the 2004 worldwide total of gas-burning plants.

Wedges 6-8 involve carbon capture and storage:

6. Capturing CO₂ at power plants. Carbon capture and storage at power plants can stop about 90% of the carbon from reaching the atmosphere, so we can get a wedge by doing this for 800 GW of coal-burning power plants or 1600 GW of gas-burning power plants by 2054. One way to do carbon capture and storage is to make hydrogen and CO₂ from fossil fuels, burn the hydrogen in a power plant, and inject the CO₂ into the ground. So, from one viewpoint, building a wedge’s worth of carbon capture and storage would resemble a tenfold expansion of the plants that were manufacturing hydrogen in 2004. But it would also require multiplying by 100 the amount of CO₂ injected into the ground.

7. Capturing CO₂ at plants that make hydrogen for fuel. You’ve probably heard people dream of a hydrogen economy. But it takes energy to make hydrogen. One way is to copy wedge 6, but then ship the hydrogen off for use as fuel instead of burning it to make electricity at power plants. To capture a wedge’s worth of carbon this way, we’d have to make 250 megatons of hydrogen per year from coal, or 500 megatons per year from natural gas. This would require a substantial scale-up from the 2004 total of 40 megatons of hydrogen manufactured by all methods. There would also be the task of building the infrastructure for a hydrogen economy. The challenge of injecting CO₂ into the ground would be the same as in wedge 6.

8. Capturing CO₂ at plants that turn coal into synthetic fuels. As the world starts running out of oil, people may start turning coal into synfuels, via a process called coal liquefaction. Of course burning these synfuels will release carbon. But suppose only half of the carbon entering a synfuels plant leaves as fuel, while the other half can be captured as CO₂ and injected underground. Then we can capture a wedge’s worth of CO₂ from coal synfuels plants that produce 1.8 teraliters of synfuels per year. For comparison, total yearly world oil production in 2004 was 4.7 teraliters.

Now: What are the pros and cons of these four wedges? What is the biggest thing that Pacala and Socolow overlooked?

I’m puzzled about the last wedge. Pacala and Socolow say 1 gigaton carbon/year is the flow of carbon in 24 million barrels/day, or 1.4 teraliters/year. They assume the same value for synfuels and allow for imperfect capture, which leads them to conclude that carbon capture at synfuels plants producing 1.8 teraliters/year of synfuel can catch 1 GtC/year. But this calculation doesn’t make sense to me. If we’re catching just half the carbon, and 1 GtC/year = 1.4 teraliters oil/year, don’t we need to generate at least twice that — 2.8 teraliters synfuel/year — to catch wedge’s worth of carbon?

I’m also unclear what percentage of the carbon you can actually capture while turning coal into synfuels. Can you really capture half of it?

There’s also another funny feature of this last wedge. If we assume people are already committed to making synfuels from coal, then I guess it’s true, they’ll emit less carbon if they use carbon capture and storage as part of the manufacturing process. But compared to making electricity or hydrogen as in wedges 6 and 7, turning coal into synfuels seems bound to emit more carbon, even with the help of carbon capture and storage.

In general, it only makes sense to talk about how much carbon emission some action prevents when we compare it to some alternative action. That’s pretty obvious, but it gets a bit confusing when some of Pacala and Socolow’s wedges look like plausible alternatives to other ones.

Another question: how does carbon capture and storage work, actually? Summarizing Pacala and Socolow, I wrote:

One way to do carbon capture and storage is to make hydrogen and CO₂ from fossil fuels, burn the hydrogen in a power plant, and inject the CO₂ into the ground.

But I’d like to learn the details!

Carbon Emissions in 2009

A news item relayed to us from David Roberts:

• Richard Black, 2009 carbon emissions fall smaller than expected, BBC News, 21 November 2010.

“What we find is a drop in emissions from fossil fuels in 2009 of 1.3%, which is not dramatic,” said lead researcher Pierre Friedlingstein from the UK’s University of Exeter.

“Based on GDP projections last year, we were expecting much more.

“If you think about it, it’s like four days’ worth of emissions; it’s peanuts,” he told BBC News.

The headline figure masked big differences between trends in different groups of countries.

Broadly, developed nations saw emissions fall – Japan fell by 11.8%, the UK by 8.6%, and Germany by 7% – whereas they continued to rise in developing countries with significant industrial output.

China’s emissions grew by 8%, and India’s by 6.2% – connected to the fact that during the recession, it was the industrialised world that really felt the pinch.

The news story is based on this article, which is apparently not freely available:

• P. Friedlingstein et al., Update on CO₂ emissions Nature Geoscience, 21 November 2010.

By the way: how come I can afford to create a link to the original article, while the BBC and other mass media cannot? Is it really so bloody difficult? Isn’t it just basic good journalism?

Also by the way: I really like getting good suggestions for environmental news stories to blog about… but I love it when people join the Azimuth Forum and post links to these news articles under News and Information.

Some puzzles. Guess before you google:

1) Which nation has the highest carbon emissions per person? In 2007 its per capita carbon emissions were almost 3 times that of the USA. I bet it’s still the champion today.

2) Say I make some round-trip flights from Los Angeles to Singapore, with one stop each way. How many flights would it take to burn as much carbon as an average US citizen does in a year? A rough estimate, please!

3) How many such flights would equal the yearly carbon emissions of an average world citizen?

(I am calculating the footprint of a flight using Terrapass. I have no idea how accurate it is or how it works. Also: all my figures only count carbon emissions from burning fossil fuels.)

Fossil Fuel Subsidies

Today my friend Bruce Smith pointed out this:

• How to save $300 billion, The Economist Online, 12 November 2010.

Here’s an executive summary for you busy folks:

This year’s World Energy Outlook, put out by the International Energy Agency, says that governments spent $312 billion on subsidies for fossil fuels in 2009.

According to the IEA, eliminating these subsidies would reduce the world’s energy demand by 5%: the current energy consumption of Japan, Korea and New Zealand combined. It would also reduce carbon emissions by about 0.4-0.5 gigatons by 2020.

(I think they mean annual carbon emissions. They also say “this is more than a third of the difference between business-as-usual emissions and the level needed to stand something like a 50:50 chance of limiting global warming to two degrees centigrade”, but that seems overly optimistic to me, given the figures I’ve been reading.)

Of the $312 billion in subsidies, more than a fifth comes from one country: Iran. To keep fuel prices as low as ten American cents a liter for gasoline — two cents for diesel — the Iranian government spent $66 billion in 2009. That’s $895 per person, or 20% of their GDP.

Saudi Arabia’s subsidy is even higher per capita, though lower overall and under 10% of GDP. (Guess what percent of their GDP comes from oil.)

Uzbekistan’s fossil fuel subsidy is even more absurd: a whopping 32% of GDP.

What is it for the USA, and the other countries the readers of this blog live in? What can we do to reduce or end these obscene subsidies?

The Azolla Event

My friend Bruce Smith just pointed out something I’d never heard of:

• Azolla event, Wikipedia.

As you may recall, the dinosaurs were wiped out by an asteroid about 65 million years ago. Then came the Cenozoic Era: first the Paleocene, then the Eocene, and so on. Back in those days, the Earth was very warm compared to now:

Paleontologists call the peak of high temperatures the “Eocene optimum”. Back then, it was about 12 °C warmer on average. The polar regions were much warmer than today, perhaps as mild as the modern-day Pacific Northwest. In fact, giant turtles and alligators thrived north of the Arctic circle!

(“Optimum?” Yes: as if the arguments over global warming weren’t confusing enough already, paleontologists use the term “optimum” for any peak of high temperatures. I think that’s a bit silly. If you were a turtle north of the Arctic circle, it was indeed jolly optimal. But what matters now is not that certain temperature levels are inherently good or bad, but that the temperature is increasing too fast for life to easily adapt.)

Why did it get colder? This is a fascinating and important puzzle. And here’s one puzzle piece I’d never heard about. I don’t know how widely accepted this story is, but here’s how it goes:

In the early Eocene, the Arctic Ocean was almost entirely surrounded by land:

A surface layer of less salty water formed from inflowing rivers, and around 49 million years ago, vast blooms of freshwater fern Azolla began to grow in the Arctic Ocean. Apparently this stuff grows like crazy. And as bits of it died, it sank to the sea floor. This went on for about 800,000 years, and formed a layer 8 up to meters thick. And some scientists speculate that this process sucked up enough carbon dioxide to significantly chill the planet. Some say CO₂ concentrations fell from 3500 ppm in the early Eocene to 650 ppm at around the time of this event!

I don’t understand much about this — I just wanted to mention it. After all, right now people are thinking about fertilizing the ocean to artificially create blooms of phytoplankton that’ll soak up CO₂ and fall to the ocean floor. But if you want to read a well-informed blog article on this topic, try:

• Ole Nielsen, The Azolla event (dramatic bloom 49 million years ago).

By the way, there’s a nice graph of carbon dioxide concentrations here… inferred from boron isotope measurements:

• P. N. Pearson and M. R. Palmer, Atmospheric carbon dioxide concentrations over the past 60 million years, Nature 406 (6797): 695–699.

Stabilization Wedges (Part 1)

Okay, let’s look at some plans for combating global warming! And let’s start with this paper:

• Stephen Pacala and Robert Socolow, Stabilization wedges: solving the climate problem for the next 50 years using current technologies, Science 305 (2004), 968-972.

I won’t try to summarize it all today, just a bit.

Stephen Pacala and Robert Socolow wrote this now-famous paper in 2004. Back then we were emitting about 6.2 gigatons of carbon per year, there were 375 ppm of carbon dioxide in the atmosphere, and many proposals to limit global warming urged that we keep the concentration below 500 ppm. Their paper outlined some strategies for keeping it below 500 ppm.

They estimated that to do this, it would be enough to hold emissions flat at 7 gigatons of carbon per year for 50 years, and then lower it to nothing. On the other hand, in a “business as usual” scenario, they estimate the emissions would ramp up to 14 gigatons per year by 2054. That’s 7 too many.

So, to keep emissions flat it would be enough to find 7 ways to reduce carbon emissions, each one of which ramps up linearly to the point of reducing carbon emissions by 1 gigaton/year in 2054. They called these stabilization wedges, because if you graph them, they look like wedges:

Their paper listed 15 possible stabilization wedges, each one with the potential to reduce carbon emissions by 1 gigaton/year by 2054. This is a nice way to start thinking about a very big problem, so many people have adopted it and modified it and criticized it in various ways, which I hope to discuss later. Right now I’ll only tell you about the original paper.

But before I even list any of their stabilization wedges, I should emphasize: stabilizing emissions at 7 gigatons is not enough to stay below 500 ppm forever! Carbon dioxide stays in the atmosphere a very long time. So, as Pacala and Socolow note:

Stabilization at any level requires that net emissions do not simply remain constant, but eventually drop to zero. For example, in one simple model that begins with the stabilization triangle but looks beyond 2054, 500-ppm stabilization is achieved by 50 years of flat emissions, followed by a linear decline of about two-thirds in the following 50 years, and a very slow decline thereafter that matches the declining ocean sink. To develop the revolutionary technologies required for such large emissions reductions in the second half of the century, enhanced research and development would have to begin immediately.

What’s the “declining ocean sink”? Right now the ocean is absorbing a lot of CO₂, temporarily saving us from the full brunt of our carbon emissions — while coral reefs, shellfish and certain forms of plankton suffer from increased acidity. But this won’t go on forever; the ocean has limited capacity.

Pacala and Socolow consider several categories of climate wedges:

• efficiency and conservation
• shifting from coal to gas
• carbon capture and storage
• nuclear fission
• renewable energy sources
• forests and agriculture

Today let me just go through the first category. Here they list four wedges:

1. Efficient vehicles: increase the fuel economy for 2 billion cars from 30 to 60 miles per gallon. Or, for those of you who don’t have the incredible good luck of living in the USA: increasing it from 13 to 26 kilometers per liter. When they wrote their paper, there were 500 million cars on the planet. They expected that by 2054 this number would quadruple. When they wrote their paper, average fuel efficiency was 13 kilometers/liter. To achieve this wedge, we’d need that to double.

2. Reduced use of vehicles: decrease car travel for 2 billion 30-mpg cars from 10,000 to 5000 miles per year. In other words: decreasing the average travel from 16,000 to 8000 kilometers per year. (Clearly this wedge and the previous one are not additive: if we do them both, we don’t save 2 gigatons of carbon per year.)

3. Efficient buildings: cut carbon emissions by one-fourth in buildings and appliances. This could be done by following “known and established approaches” to energy efficient space heating and cooling, water heating, lighting, and refrigeration. Half the potential savings are in the buildings in developing countries.

4. Efficient coal plants: raise the efficiency of coal power plants to 60%. In 2004, when they wrote their paper, “coal plants, operating on average at 32% efficiency, produced about one fourth of all carbon emissions: 1.7 GtC/year out of 6.2 GtC/year.” They expected coal power plants to double their output by 2054. To achieve this wedge, we’d need their average efficiency to reach 60%.

There are lot of questions to ask! Which do you think are the most easily achieved of these wedges? What are the biggest problems with their reasoning so far? And so on…

I would love any interesting information you have on: 1) ways to make vehicles more efficient, 2) ways to coax people to drive less, 3) ways to make buildings more efficient, and 4) ways to make coal power plants more efficient. Please post it here, with references if you can!

I’ll conclude for now with a couple of tiny points. First, they seem to vacillate a bit between saying there were 6.2 and 7 gigatons of carbon emitted in 2004, which is a bit odd, but perhaps just a way of giving the world a bit of slack before levelling off emissions at 7 GtC/year. I guess it’s not really a big deal.

Second, they aren’t idiots: despite the above graph, they don’t really think carbon emissions will increase linearly in a business-as-usual scenario. This is just a deliberate simplification on their part. They also show this supposedly more accurate graph:

They say the top curve is “a representative business as usual emissions path” for global carbon emissions in the form of CO₂ from fossil fuel combustion and cement manufacture, assuming 1.5% per year growth starting from 7.0 GtC/year in 2004. Note this ignores carbon emissions from deforestation, other greenhouse gases, etc. This curve is growing exponentially, not linearly.

Similarly, the bottom curve isn’t flat: it slopes down. They say the bottom curve is a “CO₂ emissions path consistent with atmospheric CO₂ stabilization at 500 ppm by 2125 akin to the Wigley, Richels, and Edmonds (WRE) family of stabilization curves described in [11], modified as described in Section 1 of the SOM text.”

Here reference [11] is:

• T. M. L. Wigley, in The Carbon Cycle, eds. T. M. L. Wigley and D. S. Schimel, Cambridge U. Press, Cambridge, 2000, pp. 258–276.

and the “SOM text” is the supporting online material for their paper, which unfortunately doesn’t seem to be available for free.