Carbon Catch-and-Release

Around 15 years ago, my father Steve and I were browsing in T. J. Maxx (which I'll forever think of as "Temporal Jail: Maximum Security" thanks to Jasper Fforde) and came across a soda siphon, aka the seltzer bottles that seem to only appear in old movies for comedians to spray each other with. We decided to buy one, and we've continued using them to this day (after spraying each other with it a couple times, naturally).

If you've ever seen one used, the force of the water when it sprays out is impressive, and I was curious if I could derive how fast it's moving. In looking for an answer, I found the Hagen-Poiseuille equation, which relates a change in pressure to the flow of a liquid:

where μ is the viscosity of the water, L the length of the pipe, Vdot the volume flow rate, and r the radius of the pipe. The pressure difference is simply the pressure we add when we charge up the bottle. The CO2 cylinders that go in the bottle (yes, we buy 100 at a time) hold 8 grams of gas each, to charge 1 liter of water. To find the pressure that results in, we can use the Ideal Gas Law:

where n is the number of moles of gas, R is a constant, and T is the temperature. We can get the number of moles from the molar mass of carbon and oxygen. Putting everything together, and solving the differential equation gives

where Vtot is the total volume of the bottle, and V0 is the initial amount of water. I tried plotting this, but I was having some trouble with the units – I think I missed a factor of 1000 somewhere between the millipascals, (kilo)grams, and centimeters. I wound up fudging it to get more reasonable results:

This says that the bottle will be empty in about 8 seconds (and probably not 8 milliseconds, as I initially calculated). For the velocity we get

For reference, 1 cm/s at the top of this graph is about 2 feet every minute, which is way too slow, but leaving in the factor of 1000 is way too fast! I'm not sure where I went wrong with this, but it may be that the Hagen-Poiseuille equation is not the right one to use here – Wikipedia says it "describes the pressure drop due to the viscosity of the fluid", and so it may not apply to the flow due to the difference in pressure. Oh, well – Another of Nate's "that didn't work!" posts.

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!