Mar 11 2008
Bathtub Analogy: Why the Global Temperature Giggles
JohnMashey posted this at Skeptical Science.
A bathtub is being filled [sun], slightly faster than it is being drained [heat radiation]. You have a few floats, measuring the depth of the water. The depth would go up smoothly, except there’s a kid splashing around in the bath.
Sometimes the kid lies back in the water, in which case the overall water level goes up [El Nino], but with waves, so that some floats go down.
Sometimes the kid sits up, in which case the overall water level temporarily goes down [La Nina], but with waves, so a few of the floats go up.The kid splashes around the whole time, jiggling all floats second by second.
At any point in time, there is a certain amount of water, but the average as measured by 1% of the floats is subject to lots of jiggles.
Still, the water *is* going up, as long as more as coming in than draining out, and the physics of GHGs say that we’re slowly plugging the drain.
(No Ratings Yet)
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24 Responses to “Bathtub Analogy: Why the Global Temperature Giggles”
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Good one. John Mashey emphasizes the small fluctuations vs. the trend; to extend the bathtub analogy to mitigation incentives, take a look at Michael Tobis’ version, starring a bathtub and an economist: http://initforthegold.blogspot.com/2007/06/economics-and-global-change-bathub.html .
I invite an experiment.
Start filling a bath with a partially open drain. Adjust the inflow so it just exceeds the outflow. Get it asjusted before there is more than a couple of inches of water in the bottom. Put in the floats if you wish.. Skip the kid.
Watch and wait.
The bathtub will never fill.
(which may merely indicate that no analogy is perfect, or may indicate something more profound)
[Reply: Further assume that there is only one source for the water (the faucet), and one sink (the drain). Evaporation is negligible.]
Okay, that’s intriguing – I’ll guess that the flow through the drain increases as the tub fills, because of increased water pressure/weight? It’s sort of an Iris hypothesis for baths…
Ian–
Yep. The outflow is predicted by applying Bernouilli’s equation from the surface of the water to the top of the drain, and then something called “the mechanical energy” equation from the entry of the drain to the exit of the drain. (The mechanical energy equation looks a lot like Bernouilli’s equation, but accounts for frictional losses.)
The solution I just described assumes the water level is sufficiently high and/or the drain rate sufficiently low to permit you to find a point on the surface of the water that’s not sucked down by the vortex that may form when draining, and then draw a line from that point to the drain, without passing through regions with high shear. But, other than that, it’s a typical undergraduate problem done second semester sophomore year or third semester junior year in college. (Which year depends on the discipline.)
John Lederer’s example uses different assumptions than mine, and of course, is not generally correct, although there are combinations of paramters for which it is. For 2″ of water, in a 12″ bath, with an infloiw just barely greater than outflow, and no clogging of the drain, the tub indeed will never fill much beyond 2″, because the outflow rate is roughly proportional to the pressured/depth.
With a 12″ bath, 10″ level (including kid), filling at 1″/minute, and enough clogging to keep it at that rate, it’s a different story, and will overflow even without the clogging.
In the skepticalscience article, my discussion started with the Hansen, Nazarenko, et al paper on Earth’s energy imbalance, which is clearly more than “just exceeds”, and used this as a simple analogy. The discussion had started elsewhwere with folks who didn’t like to hear about First & Second Laws of Thermodynamics and radiative energy balances. Anyway, see Figure 2, which sure looks like there’s a noticable and continuing imbalance, i.e. ,the clogging is keeping up with any pressure increase.
Anyway, nothing profound. Analogies only go so far, but my version actually had enough detail to match Figure 2, whereas John’s sets the clogging to zero.
John Mashey–
The analogy is useful, and I like the way you use it.
But it’s a bit hilarious in the “Playing Tetris while the bathtub overflows” example at another blog. With every bathtub I’ve ever owned, the drain is large enough that I can’t fill it while the drain is open, even if the water is running as fast as possible.
One problem with analogies is that sometimes, you need to be careful they don’t seem to prove the case you intend to disprove!
We’re talking about real tubs now?
The drain in my tub is partially clogged so that when I take a shower about 2.5cm of water is in the tub by the time I finish. Drano doesn’t work. I should probably call a plumber.
I think it depends on the type of plug. I’d guess the ones with manual stoppers [which are basically binary], it is indeed pretty hard to overfill. Ours are the analog sort, where would normally be open or closed, but actually, if one is careless, and doesn’t turn the control all the way, one can actually get the intermediate points. Even with full-open, the are of the opening is a ring, not he full circle.
This discussion brings these results to mind:
“Understanding Public Complacency About Climate Change: Adults’ mental models of climate change violate conservation of matter“
Steve: yes, I saw that a while back, and the idea must have stuck in my mind, althouhg of coruse, various people have used some flavor of bathtub analogy. Of course, the main point in my use of it was the fact that the jiggles of the floats didn’t really mean much in the short term compared to watching the overall level. I still wouldn’t call it the best analogy, as it needs things like a long lag time for ever cleaning out the clog. maybe atmoz can find an atmosphere plumber or GHG-Drano
Atmoz-
Yes. Get a snake. That should clear it up. When my brothers-in-law rented a place the once found two tonka truck’s that had been flushed down by the previous tenants. Once those were out, the tub drained like magic!
I have to admit, I had to make a youtube video. It’s my second. This is how a sink should drain:
Sink draining!
Well, at the risk of flogging my example to a froth in order to try to make something trivial profound.
I suggest that if all you have to measure by are the floats bobbing in the unpredictable waves caused by the restless child, which being a scientist, you carefully measure chart and then calculate 182 trend lines from, apply curve smoothing algorithms invented by mathematicians trying to explain to their wives why it was OK to come home blotto, and adjust all by means of the proxy data of the child’s giggles:
1. If you expected the water level to stabilize, it will.
2. If you expected the water level to keep rising, it will.
You will, of course, also conclude that it is obvious that further funding is necessary to study the problem more fully.
[...] tip: Atmoz, where one can find engaging [...]
John M–
I think the analogy is good when limited to explaining why there are jiggles. The only difficulty is that, with respect to the discussion of what’s really happening on the earth, the analogy itself ends up containing all the same question:
How fast is water added?
How fast does the water drain? (In radiative physics, the rate of heat loss depends on the planet temperature. In tubs, the flow out the drain depends on the height of the water.)
Are there any fail-safe mechanisms? (My tub and bathroom sinks have little overflow valves at the top. My kitchen sink does not.)
Can we diagnose the trend by measuring the surface level? Are we doing it right? How long will it take to detect any trend?
So, in some sense, the tub analogy is perfect! For explaining jiggles, it’s great. For explaining the full range of what might be happening, it’s great.
It just fails if it’s supposed to prove to us that we must go turn the water off now. (I don’t think that’s the point you wanted to make– it seemed to be the point on another blog.)
But, I am thinking, a Lava Lamp analogy might be fun. Maybe I could find a Lava Lamp and hook it up to something to make increase the power addition linearly? And then slap a thermistor on the interior surface? This could be messy, and dangerous.
But I’m an engineer. Messy and dangerous is no problem. (The difficulty is I’m cheap. So, I need to find the Lava Lamp at Goodwill!)
I think the tetris-playing economist is an interesting analogy, and I don’t take it as arguing that we “must” shut off the bath now. For me, the argument would be that it’s good to make the effort to _check_ the bath’s water level, which may in fact be getting too high, rather than refusing to check. (In Tobis’ analogy, you need to turn the water off immediately only if it’s just at the point of overflowing.)
The crucial point of the tetris player analogy, in my interpretation, is about discount rates for the future: we place more value on an option in the near than in the far future. The rate of discounting and the shape of the discount curve are important to setting current policies and expenditures. (Most of the criticisms of the Stern Report in the UK seemed to be about his choice of discount rate.) The neat twist in the tetris example is that the player sees the trade-off between the current and future utilities (that is, the pleasure of playing now and the pain of fixing the house later), but has construed the situation so that he won’t collect the information that he knows he needs to make a decision.
You can complicate it into a bit of game theory, too: suppose the player is engaged in a tetris competition with his roommate, and each player is on a different computer playing simultaneous head-to-head games. Leaving the game to check the bath gives the advantage in tetris to the competition; so the incentives for each player favor waiting for the _other_ player to stop playing first to check/turn off the bath
Lucia:
1) Yes, there are several different bathtub analogies bubbling around.
2) Our tubs have overflow outlets at the top, but if the drain is closed or blocked enough, with the tap running all-out, they are insufficient to stop an overflow.
3) How fast does it drain: it’s not just pressure (temperature)
- it’s certainly related to the effective size of the outlet, including clogging
- and there are certainly other fluid-flow perturbations, probably minor here, but that perhaps are akin to albedo changes, sulfate aerosol reflectance, i.e., kid sticks a limb under the flow and some water goes on the floor, not into the tub.
[This may be getting strained, but not having kids, I can only theorize that such behavior might occur :-)]
4) In any case, for everybody, I recommend going back to the original post http://www.skepticalscience.com/news.php?n=24#417, whose *first* half talked about the Hansen, Nazarenko, et al paper “”Earth’s Energy Imbalance: Confitmation and Implications”, Science, 2005. http://pubs.giss.nasa.gov/docs/2005/2005_Hansen_etal_1.pdf
The confirmation of the ocean heat content rise is illustrated in Fig 2, which is where I got “The water *is* going up.” Although getting strained further, this would be the equivalent of having time-averaging pressure sensors in the bottom of the tub, which presumably would be less sensitive than the instantaneous jiggles on the surface.
Anyway, that article is well worth taking the time to digest. Among other things, it talks about the warming “in the pipeline” even if current GHG concentrations were to be constant (about .6C). You may note that in my use of the bathtub, unlike MT’s, I said nothing about turning the faucet (sun) off, although of course, people have proposed things like artificial volcanoes or orbiting mirrors to raise the albedo enough to give us breathing time to unplug the drain some. The closest I’ve gotten to that is once whimsically back-of-the-envelope estimating that 3 years of the world’s total aluminum production converted to aluminum foil would let us cover Greenland, but I failed to pursue that further
The big issue *isn’t* turning off the faucet [which we don't know how to do], it’s adjusting/unclogging the drain, which takes serious work, and even if you stopped clogging the drain right now (GHG flat, which we’re not close to doing), the water (temperature) would still rise for a while. ["25 to 50 years are needed for Earth's surface temperature to reach 60% of its equilibrium response."]
The “equilibrium response” is what John Lederer alluded to in his first post, i.e., that for appropriate sets of conditions, the water rises, but slower and slower, approaching an asymptote. where the increased pressure raises the outflow enough to balance. Of course, as conditions change, the asymptote changes, and clogging the drain raises the eventual level even higher, and we’re continuing to clog the drain more, not less, so the effective aperture keeps reducing in size.
If only it were as simple as turning off the faucet, but it isn’t.
John M:
I guess you could suggest that the key insight from the “Tetris playing tub filler” is that the Tetris-playing-bathtub-man doens’t monitor the problem. But if that’s the insight, that analogy to AGW is really, truly inapt.
With regard to AGW, deniers, warmers, skeptics, alamist or what have you, everyone is feverishly monitoring the temperatures, CO2, GHG’s, cosmic rays, the sun, the PDO, ENSO , tree rings, boreholes, Vostok ice cores& etc. I’m surprised someone isn’t reading Nostradamus to tell us what he said. In the true “tub to AGW” analogy, the bathroom would be so chock full of people standing around taking measurements, checking measurements and arguing over the meaning of tub surface level measurents. They are also arguing about whether or not the drain is clogged, or whether the towns water pressure might not have natural long term cyclical variations.
They may not be fixing the problem, but they sure are monitoring it!
So, in other words, if you are correct about the “key” insight, who plays the role of the Tetris-bath-tub filler, the only person who has decided to not monitor the tub level? (Maybe Mother nature who is doing the laundry or washing dishes and causing the naturaly cyclic variations by using water?)
But, I’m afraid that the “Tetris playing economics” problem is sort of silly. The economic questions in the Tetris problem are: Why did Tetris-man start filling the tub if he didn’t want to take a bath? If Tetris-man likes to start filling bathtubs and then play Tetris, why doesn’t he bring the gameboy into the bathroom? And if his tub doesn’t drain, why doesn’t he hire a plumber? Presumably, his next bath shouldn’t be that far off into the future?
Getting back to real bathtubs: Yes. I understand John L and you are discussing different gedunkan experiments. In the limit that the tub drains slowly and is nearly full when water draining and entering are nearly at equilibrium, the height of the water varies as:
h= C (Q/ Area) ** 2 where “Area” is the cross-sectional area of the drain, Q is the flow rate and “C” is a constant. (C is equal to the product of gravity and so coefficients that depend on the geometric features of the drain. Some odd bathtub geometries could screw this up, and in particular, it will break down if the equilibrium level of the tub is shallow. But for most bathtub problems, this would work ok. It would work even better if the spout weren’t directly over he drain!)
So, yes, John L’s analogy is to assume “A” is constant, and we just nudge Q some constant amount then watch “H” increase a bit. In your analogy, “Area” is constantly decreasing over time. So, h increases, and never reaches a steady state because “Area” is constantly decreasing.
I interpret the bathtub as the earth with certain boundaries where water (heat radiation) escapes when beyond those boundaries. With limits in mind, I have a question.
I understand that former NASA researcher Miskolczi has written in “Greenhouse effect in semi-transparent planetary atmospheres” at:
http://met.hu/doc/idojaras/vol111001_01.pdf
that 1) generally accepted climate models based upon semi-infinite atmospheric parameters introduce for clear or optically thin atmospheres large errors into equilibrium surface temperatures, 2) finite semi-transparent atmospheric models, on the other hand, give the “correct relationships between the fluxes, greenhouse parameters and the flux optical depths…good enough to give quantitative estimates with reasonable accuracy”, and 3) the latter models resolve the surface temperature discontinuity problem, and show “significantly reduced greenhouse effect sensitivity to optical depth perturbations”.
I read the October, 2006 “peer reviewed” paper published in Quarterly Journal of the Hungarian Meteorological Service as a possible explanation for the findings reported by Lindzen at:
http://www.ecoworld.com/blog/2008/03/12/co2-global-warming.
The findings by Miskolczi were rejected by the IPCC because they were unsupported by the literature, but you would expect nil written support since he questioned the conventional approach in the first instance. If his findings are nevertheless valid and meaningful, then the rejection certainly appears arbitrary and capricious.
Has anyone given the foregoing more attention and, if so, what conclusion(s)? Are climate models presently based upon assumptions of atmospheres without boundaries and, if so, do they make sense?
lucia,
You seem eager to reject the tetris-bathtub analogy. No problem, nothing hinges on the analogy itself. But I would suggest that the animating economic point is assessing what, and how much, to do now in anticipation of future climate states.
re: Miskolczi
See #265 in:
http://tierneylab.blogs.nytimes.com/2008/03/06/global-warming-payola/
The Miskolczi paper is 40 pages crammed with equations. I’ve looked at it. I haven’t *read* it, I suspect most people quoting what it’s supposed to mean haven’t either, or couldn’t. If it were true, it’s hard to see how ice-age cycles happen, which probably means there’s an error in there somewhere.
Right now, the blogosphere’s two favorite papers that “prove” AGW is wrong seem to be the Miskolczi paper and the Richard Mackey paper about Rhodes Fairbridge.:
Google: Miskolczi greenhouse
and
Google: richard mackey rhodes fairbridge
I haven’t done a study on the propagation of these, but i suspect it won’t be too different than what I saw in the silly Monckton / Schulte / Oreskes thing last summer, although that was even dumber. The NHS endocrine surgeon’s study was supposed to have demolished Oreskes was totally incompetent, and didn’t even get published in Energy & Environment. Nevertheless, it was propagated widely. Sigh.
Miskolczi at least looks like someone real, but peer review doesn’t catch every mistake. There are probably some grad students somewhere starting to look at it.
John Mashey:
Thank you for commenting as you did here and at the New York Times blog (which comment I missed).
To your first, most basic question at the other blog, “Can you, from reading the paper, which is 40 pages of dense equations, tell me what it really means, and how it really blows up the models?”, I answer “No!”. I’m a spectator in the global warming science debate with a present appetite to read as much on the subject (while skimming the dense equations) as my solo practice will permit and with occasional urge to ask questions.
Given that Miskolczi is reported to be an atmospheric physicist, his final paper is dated October 29, 2006, he reportedly left NASA under “a cloud”, he was apparently given terse treatment by the IPCC, and that I had not read about his work until recently, my curiosity was piqued.
I believe now the phrase “runaway greenhouse” also mentioned by Miskolczi has been overemphasized in the characterization of his attempt to explain his view of an Earth system more efficient in radiating heat than present models are said to allow. Honestly, I prefer that view to have substance and that a conclusion about it be reached, certainly sooner than waiting for conclusive trends to appear. After all, we are presently witnessing under the alternative a published conclusion that near zero CO2 emissions are required to stabilize temperatures, Lovelock’s “optimism” that, if lucky, we have “20 years before it hits the fan”, and the storage of seeds in Norwegian “Doomsday Vaults”.
Ian:
I agre that this is an important economic point.
I just think the Tetris - bathtub analogy would have served the “rebuttal” side to Tobis’s favored position better.
Unfortunately, poor analogies are worse than no analogies.
The only problem’s I see with the bathtub analogy (which I love by the way), is that we, as humans, somehow assume was can either a) turn off the “water”, or b) open/close the drain.
Well, we are narrowing the drain already, and it’ hard to find any scenario where we don’t keep doing that a while.
On the other hand, maybe we can use geoengineering to get the kid:
a) To stick a foot under the tap, defelcting some of the water outside the tub before it evern enters [i.e., people have proposed genreating extra sulfates to reflect sunlight].
OR
b) Splash a lot, so that some of the water goes oever the edge. This is sort of like raising the albedo at the surface by covering Norhter canda with aluminum foil