Issue 058 - Climate - Mass-to-volume conversion
How much sea-level rise comes from 12.5 trillion tons of ice?
A new satellite-based analysis estimates that Greenland and Antarctica lost about 12.5 trillion U.S. tons of ice between 1979 and 2023 as their ice sheets lost mass.
The problem
Estimate how much global average sea level would rise if that amount of land ice entered the world's oceans.
Build the estimate from the ice mass rather than using a remembered sea-level conversion.
You will need to make your own assumptions about the density of ice or liquid water, the size and surface area of Earth, what fraction of Earth's surface is ocean, and how uniformly the added water can be treated as spreading across the oceans.
Express your final answer in millimeters, centimeters, or meters of global average sea-level rise.
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
First, I assumed the article headline is not talking about metric tons, so I would need to multiply 12.5 trillion tons by about 900 kg to find the mass of melt in kg so units are easier to work with. That gives about 1.1 x 10^16 kg.
ice mass
~= 12.5 x 10^12 U.S. tons x 9 x 10^2 kg/ton
~= 1.1 x 10^16 kgNext, assuming the volume of melted water added to the oceans is about 1000 kg/m3, divide that mass by 10^3 to get cubic meters of water. That gives about 11 x 10^12 m3 of water.
meltwater volume
~= 1.1 x 10^16 kg / 10^3 kg/m3
~= 1.1 x 10^13 m3Next, I know the surface area of Earth is about 5 x 10^14 m2, and I think the oceans cover about 70% of the surface area, so 5 x 10^14 m2 x 0.7 gives about 3.5 x 10^14 m2 of ocean coverage.
ocean area
~= 5 x 10^14 m2 x 0.7
~= 3.5 x 10^14 m2Assuming the 11 x 10^12 m3 of water is distributed evenly across all the oceans, I would divide that by 3.5 x 10^14 m2 of ocean area, and that would give a surface level change of about 3 x 10^-2 m, or about 3 cm of change.
sea-level rise
~= 1.1 x 10^13 m3 / 3.5 x 10^14 m2
~= 3 x 10^-2 m
~= 3 cmCalibration Score
Matt's Calibration Score: 100 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 30/30. The U.S.-ton conversion, water-density peg, Earth surface area, and ocean-surface fraction were all right where they needed to be.
Model: 30/30. Converting mass to water volume and dividing by global ocean area is exactly the right first-order model for global mean sea-level equivalent.
Math: 10/10. The unit conversion and scientific-notation arithmetic were clean.
Result: 30/30. The estimate of about 3 cm is within a few percent of the reported 31.4 mm contribution.
Grounding facts
The Scientific Data paper combines 42 independent satellite-based estimates of Greenland and Antarctic ice-sheet mass balance. It reports a combined loss of 11,309 billion metric tons of ice between 1979 and 2023.
AP reports the same loss as about 12.5 trillion U.S. tons, and says the meltwater pushed global sea level up by about 3.1 cm.
NOAA gives the ocean's surface area as about 360 million km2, which is 3.6 x 10^14 m2. Water density is close to 1,000 kg/m3, so one metric ton of water is about one cubic meter.
After checking sources
Check and recalibrate
The compact sourced version is almost the same as Matt's first pass. Start with the study's metric mass estimate:
ice mass
~= 11,309 billion metric tons
~= 1.13 x 10^13 metric tons
~= 1.13 x 10^16 kgConvert that mass into liquid-water volume:
water volume
~= 1.13 x 10^16 kg / 1 x 10^3 kg/m3
~= 1.13 x 10^13 m3Then spread it across the global ocean:
global mean sea-level rise
~= 1.13 x 10^13 m3 / 3.6 x 10^14 m2
~= 3.1 x 10^-2 m
~= 31 mm
~= 3.1 cmThe best plain-English answer is: 12.5 trillion U.S. tons of land-ice loss corresponds to about 3 cm, or a little over 1 inch, of global average sea-level rise.
That is not meters, or even tens of centimeters. But the inverse lesson is the sharp one: if a few centimeters of average sea level requires more than ten trillion tons of land ice, then even small-looking changes in sea level represent enormous movement of mass around the planet.
Post-check reflection
Matt's reflection
Nailed this one. My understanding of the problem was correct, I converted to kg correctly, all my pegs were appropriate, and the rest of the math worked cleanly so I arrived at an answer only a few percent off of the calculated one. Not really any room for improvement here.
Even though the mass of the melt sounds enormous, when spread over the world's oceans it would only create a surface level rise of about an inch. That can really be interpreted two ways: it takes a ton of melt before considerable surface level rise occurs, or inversely, even tiny surface level rises mean huge amounts of ice melt.
Recommended memory peg
For sea-level-equivalent problems, remember: the oceans cover about 3.6 x 10^14 m2, so 1 mm of global average sea-level rise requires about 3.6 x 10^11 m3 of added water, or about 360 billion metric tons.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.