Selling Our Sol

Marika and I are up in Michigan right now for holiday visiting. Along the way we passed a large solar farm. On a previous trip, passing the same spot, Marika posed an interesting question: How much power could we possibly get from solar? Is it enough on its own?

There are a couple complications with this question: the broadest approximation would be to take all the power that hits the Earth from the Sun in 24 hours. That works out to about 6 million terrawatt-hours. The total daily use of the planet is around 459 TWh, well below the power that's hitting us from the sun! Of course, that assumes we have perfectly efficient panels, and cover the entire planet with them, so that's not exactly realistic.

Let's consider the sort of solar cells currently available. The conversion efficiency is a bit involved, but I decided to go with 15%, based on the silicon cells mentioned in that article. We also need to consider the angle that the light hits the cell. For that, we need the position of the Sun, and the tilt of the panel. You can get setups that track the Sun as it moves through the sky, but I went with a fixed panel. In that case, the optimal angle is tilted due South (North) in the Northern (Southern) hemisphere with an angle equal to the latitude, e.g. the panel would be standing up straight at the North Pole, and flat at the equator. I took the power generated by the panel to be proportional to the cosine of the angle between the Sun and the face of the panel.

As a first test, we can look at the power generated (in Watts/square meter) over the course of a day:

Notice that since we're looking at January 1st, the brightest spot moves over the Southern Hemisphere. We can also take a look at the maximum power generated each day over the map:

Notice how the maximum moves North and South with the seasons. Now for the key question: How does this compare with current energy usage?

As you can see, our energy needs (in red) are several orders of magnitude below solar generation, even with today's inefficient panels. However, it's a bit impractical to cover the entire landmass of the Earth with solar panels, so it might be best to include some wind, hydroelectric, and geothermal power sources as well. Another option would be putting the solar collectors in space, and beaming the power back down to Earth, as in a favorite game from my childhood, SimCity 2000. Of course, that would occasionally miss, and level a few buildings with the beam...

Thanks for a great question, Marika!

GRACE is Beauty in Motion

This week, I heard two talks on the Gravity Recovery and Climate Experiment (GRACE), an experiment being developed by some of my colleagues here in Florida. The goal of the experiment is to measure variations in the mass distribution of the Earth, using a pair of orbiting satellites:

JPL

As the satellites orbit, they pass over regions of greater and smaller mass, causing them to speed up or slow down. These variations will affect the leading spacecraft first, causing the separation between them to change. We can measure these changes with a laser interferometer, just like the ones used by LIGO and LISA. This results in a detailed map of how mass is distributed over the planet:

JPL

The heaviest points (in red) tend to be in mountain ranges, like the Alps and Andes.

You might be wondering (as I did) how this relates to climate. The key is that water is dense stuff, so when it moves around, it can significantly change the pull of gravity. As snow and ice melt, the water will flow to different places. The researchers have made their data available online, so I tried putting together some code to make summary plots.

The data I linked to above records the liquid water equivalent thickness, which is the depth of water over an area that would result in the measured mass per area. It covers 2002 to 2020, giving a planet-wide measurement every month. I plotted the data on a Mollweide projection and animated it in time:

I wasn't able to make it as clean as some of the diagrams the presenters showed, but you can see some seasonal variations, particularly in the Amazon region, and you can see things get significantly redder as time goes on. Looking at a single point in the middle of the North Atlantic shows an alarming trend:

As the glaciers and ice caps melt, that water flows into the oceans, raising the levels. On top of that, warm water expands, so any kind of heat added to the oceans will increase the depth. I hope we can use tools like GRACE to learn how best to reverse trends like this, and how to emphasize how necessary it is!

See Spot. See Spot Cool.

Another question from the ever-curious Papou [paraphrased]: I've been reading about the cycle of sunspot activity, and how some believe it connects to climate change. What are your thoughts on this?

Let's start off by discussing sunspots. The sun is essentially a big ball of plasma, a state of matter that's particularly good at holding electromagnetic fields. Under certain circumstances, some magnetic field lines in the sun will bunch together, and change the flow of particles around a spot. This causes the spot to cool enough to appear dark:

Getty Images via Scientific American

In the 1890s, an astronomer couple named Annie and Edward Maunder noticed that there was an unusually low number of sunspots during the years 1645 to 1715, which came to be known as the Maunder Minimum. That period is also known as the Little Ice Age*, due to the unusually cold weather. Some people draw a link between these two, and suggest that sunspot activity is responsible for changes in global climate.

In cases like this, it's important to remember the science adage, "correlation does not imply causation." This means that just because two quantities change in a similar way, it does not mean that one causes the other. Obviously, if one thing causes another, they will be correlated, but you need more than that to prove causation: You need to explain how they're linked, and show that that explanation is backed up by data.

Sunspots actually fail on both counts: Sunspots are dark because they're cooler than the rest of the sun. The Little Ice Age, though, happened during a minimum in the number of sunspots, and tracking the temperature through time shows little correlation:

Wikipedia

While it's important to consider alternate theories to popular beliefs, it's also important to discard them when they are shown to be inaccurate. As I discussed several months ago, I am not an expert in climate science, so I defer to the vast majority who identify CO2 emissions as the leading cause.

*Nifty cultural note unrelated to Physics: Charles Dickens is largely credited with giving us our collective image of a white Christmas through his story A Christmas Carol. Christmases that include snow, however, are quite rare, and it's likely that Dickens only imagined it this way because he grew up at the end of the Little Ice Age, where they were more common.

Tri Try Tro Trombe

We've been having a heat wave the last couple weeks in Annecy, and Marika and I were discussing the lack of air conditioning here. In spite of that, some buildings remain reasonably cool. Marika, being both an observant nurse and coming from a family of engineers, noticed that most buildings here are made of concrete, rather than the wood/steel buildings we have in the US. While talking to Papou a few nights ago, we mentioned this, and he identified them as Trombe walls.

Trombe walls consist of a concrete bulk, with a layer of glass or reflective coating on the outside.

Simplified from Wikipedia

The key is that UV rays from the Sun are able to pass through the glass and get absorbed by the wall. The wall heats up, which releases infrared light, but these are reflected by the glass, and remain inside. We can see this looking at the spectral reflectance:

Aleksandra Gardecka

Ultraviolet light has wavelength around 0.4 microns, the left edge of this graph, and infrared light is anything from 0.7 microns to 1 mm, beyond the right side of the graph. You can see that in between, the glass goes from mostly transmitting, to mostly reflecting. This means that the UV light will get in, but the re-emitted IR light will be reflected back.

It isn't all about the glass though: The concrete provides a large mass to store the heat when it comes in, and acts as a buffer between the inside and outside. I had hoped to make a simulation of this, but my concept of what happens is that as the weather outside heats up in the day, and cools down at night, those changes are transmitted inside with a delay, as the concrete absorbs and re-emits the heat. That results in the inside being cooler during the day, and warmer at night.

Air conditioning will still cool things better than this method, but this has the advantage of being more eco-friendly. I didn't expect international living to inspire blog posts, but thanks to Marika's observational skills, I've learned something new!

Tennesseein' is Tennebelievin'

[Title thanks to The Simpsons]

My brother-in-law Dave Willett sent me an op-ed from the Tennessean titled "Primary cause of global warming is force of gravity". Normally I hesitate to comment on science outside my expertise, but by attributing climate change to gravity, the author has brought this into my wheelhouse.

What's dangerous about this piece is that it mixes accurate information with unsupported conclusions. I thought the best way to tackle it would be a running commentary.

Climate change throughout planet earth is occurring and is observable, measurable, provable and, most importantly, unavoidable. Through numerous empirical methods the inexorable warming trend is being monitored and documented by researchers throughout the world. Thus mankind's influence on weather patterns and global warming is minuscule compared to the colossal heat-producing forces within the earth itself.

The main thing that bugs me about this passage is "thus". It suggests that the first part is in any way connected to the second, when in fact mankind's influence is painfully clear:

Wikipedia

Each line represents a different temperature-recording technique. Notice the massive uptick during the 19th and 20th centuries, strongly correlated with the advent of industrialization. This plot alone contradicts the author's thesis: Warming from the Earth's mantle would be a constant effect, with no connections to human history.

[...]
The earth itself is a heat-generating machine and is gradually warming, as is virtually every other planet, star and asteroid in our universe. The primary cause of this increase in global temperature is purely and simply the force of gravity. Due to the ubiquitous, ever-present force of gravity, our earth is gradually and inexorably shrinking. The force of gravity at the earth's surface is 9.80 m/sec/squared and increases greatly as it is measured closer and closer to the center of the earth. Gravitational pull increases the internal pressure in the earth itself and thus increases the internal temperature.

There's a lot wrong here, but one thing that's (almost) correct: The acceleration due to gravity is 9.8 meters/second^2 at the Earth's surface. Newton's Law of Gravity states

where a is the acceleration from gravity, G is Newton's constant, r is the distance to the center of mass, and M_enc is the total mass enclosed in that radius. If you plug in the radius and mass of the Earth as a whole, you will indeed get 9.8 m/s^2. However, as you descend into the Earth, not only will r get smaller, but so will M_enc. If the mass is equally distributed in a sphere, then the mass enclosed at a certain radius will be a ratio of the volumes:

where M_tot and R are the total mass and radius of the Earth, respectively. If we plug this into the equation above, we find as you descend into the Earth, the acceleration actually decreases:

This makes sense if you imagine being in the exact center of the Earth: Which way is gravity pulling you? The Earth is symmetric around you, so there can be no preferred direction, and the acceleration must be zero. It is true that the rest of the Earth is pushing in on you, creating pressure and heat, but that brings us to the next part.

The laws of thermodynamics teach us that heat is transferred by three methods: conduction, convection and radiation. Consider the colossal heat within a few thousand feet of the earth's surface and that this heat is transferred by all three phenomena, through conduction heat is transferred to the earth's surface through the tremendous dynamic circulation of the astronomical volume of molten metal and rock. Convection currents transfer heat to the earth's surface, and the radiation of the incredibly hot geological structures also raises the temperature of our environment.

Radiation of heat is an important point for climate change. Normally, heat created by the planet is radiated into space. The excess of CO2 and other greenhouse gasses, however, reflects this heat back to the surface. From here the author goes into effects of tectonic shifts, which I won't get into, because that's not my area of expertise! 

The author emphasizes the dangers of climate change, which is a great position to have, but without understanding the causes, it's impossible to arrive at a solution. As much as I mourn the decline in society's evaluation of science, worse is the misuse of science to arrive your own desired conclusion.