How to make sanity checks

In late September, I wrote a note on whether Lake Superior still remembers the last ice age.  The answer was no (read that post for why).  But along the way I illustrated a simple sanity check that would have given the author I was responding to a heads up that he was seriously wrong.

The check I used was to compare volumes.  If something the volume of Lake Superior remembered conditions for 10,000 years, then something with 100,000 times the volume would (could/should/...) take 100,000 times as long to adjust.  The ocean is that much bigger, so would take that much longer.  Yet we know (sanity) that the ocean's circulation time is only a few hundred to a few thousand years. 

This doesn't prove that the original 10,000 year estimate was wrong.  That's not the purpose of a sanity check.  Rather, the sanity check alerts us to examine the system more carefully.  Maybe there's something fundamentally wrong about using volume for comparison, maybe there's something fundamentally wrong about what lead that author to saying 10,000 year memory for Lake Superior.  As we found out, it is the original claim of 10,000 years that was severely wrong.  (Turned out to be about 6 months.)

In the comments, though, there were some noting that my approach to sanity check wasn't right.  Or at least that I could have made a better estimate than I did.  Since I take sanity checking to be a heads up process rather than a proof, I'm not very concerned with whether I chose the most accurate (I did choose one of the simplest) method.  But it is worth its own discussion how you might make better estimates.
One route to estimates is simple size comparisons.  I did volume, but you could also easily do area, or depth.  With Lake Superior being 1 unit (we then measure by Lake Superiors), the ocean is 100,000 for volume, 3750 for area, and 25 for depth.  I've rounded all of them.  The only part that's important is that 25 is much smaller than 3750, and so on.

You could also consider some general processes.  One is the diffusion of heat.  The thing about diffusion is that if it takes 1 time period (second, day, year, ...) do diffuse 1 distance unit (whether that's 1 foot, meter, Lake Superior average depth), it takes 4 time units to diffuse 2 distance units.  Time needed is proportional to the square of distance.   T ~ d^2  If it takes over 10,000 years to diffuse heat through Lake Superior, then it takes 25^2 (25*25, 625) times as long, 6.25 million years, to diffuse heat through the depth of the ocean. 

One commentator suggested treating Lake Superior as a sphere and examine surface area to volume ratios.  There's a good physical principle involved, but first I'll digress to mention a book I've heard good things about: Consider a Spherical Cow.  (The subtitle turns out to be 'A course in environmental problem solving').  This would give you either 10 times longer to change the ocean's temperature (assuming same energy per square meter of surface area is lost) to 100 times longer (diffusion from the core of the ocean-sphere has 10 times farther to go, so takes 100 times as long).

The thing is, Lake Superior, the ocean, the atmosphere, and most large climate system components are not very 3 dimensional, making a sphere a poor model.  I have myself used the 'spherical cow' approximation, for cows, for people, and quite a few other things.  But Lake Superior (and so forth) are very much larger in 2 dimensions than in the third.  As one commenter responded to the original suggestion, Superior is a thin film, not spheroidal.  563,000 meters long, 257,000 meters wide, 147 meters average depth.  (Hank Roberts, Oct 22).  In other words, more than 1000 times long or wide than it is deep.  That's also true of the ocean, atmosphere, and the Antarctic ice sheet, among many others.  They're all much closer in shape to a piece of paper than a sphere.

Depending on which route we chose for the sanity check, then, we'd have gotten the ocean taking longer, by all routes, to adjust to climate change than Lake Superior.  10 times, 25 times, 100, 625, 3750, and up to 100,000 times longer.  But if we add 1 piece of knowledge (go look for it) -- that the ocean's adjustment time is only several hundred to a few thousand years -- we have our check that the original estimate of 10,000 years for Lake Superior was (perhaps) far, far too long.  It turned out to be 20,000 times too long, though for reasons that had nothing to do with the volume or area comparisons between the lake and the ocean.

In making these estimates, and particularly the way in which we've been going about them, we're using (if not naming) concepts from dimensional analysis, nondimensionalization, and scale analysis.  I'll be returning to these concepts more directly.  On nondimensionalization, perhaps the best-known non-dimensional number is the Mach number for speed.  It's the ratio between the object's speed and the local speed of sound.  'breaking the sound barrier' meaning to go faster than Mach 1.