Guest Post by Willis Eschenbach
Did you ever sit on a hot sand beach and dig your hand down into the sand? You don’t have to dig very far before you get to cool sand … but even though it’s nice and cool a few handwidths down, the fact that it is cool doesn’t matter at all to either the temperature of your feet or to the temperature of the air. The beach air is hot, and your feet can still get burnt, regardless of the proximity of cool sand. I’ll return to this thought in a bit.
I’ve been mulling over the various time lags in the earth’s system For example, the peak temperature during the day doesn’t occur until about three hours after noon, and the hottest months of the summer are about a month and a half after the summer solstice. This is because it takes time for the heat to warm the earth, and that heat comes back out of the earth during the times in the temperature cycles when there is less forcing. I looked at that, and I thought, hmmm … a three-hour lag in a 24 hour daily temperature cycle is about an eighth of the cycle. And a month and a half lag in the annual temperature fluctuations is about and eighth of a cycle … hmmm. I wondered if they were connected.
So I pulled out my bible, Rudolph Geiger’s much-updated 1927 classic, “The Climate Near The Ground” (Amazon, ninety bucks, yikes!). [UPDATE The commenter ShrNfr notes in the comments that there are used versions of The Climate Near the Ground at Abe Books for prices under $10 … many thanks.] It is a marvelous book, from a time when people actually measured things and thought about them. I have a hard copy, it’s my main climate squeeze. However, while writing this I just noticed that an older edition is available as a FREE DOWNLOAD! (Warning: 23 Mb file, lots of pages of good stuff.) The first edition was in 1927 in German, then a second edition updated in 1941 and translated into English. Harvard University Press published the third edition in 1950, followed by a fourth edition in 1960. All of these were updated by the author. A fifth edition was published in 1995, updated by Aron and Todhunter in honor of the 100th anniversary of Geiger’s birth. The hard copy I have is the sixth edition, 2003. I see the online copy is the 1950 Harvard University version. Get it, either in hardcopy or for free. Read it. Every page is packed with actual experimental results and measurements, real science.
In both the 1950 and the modern versions there is a lovely graph showing what are called “tautochrones” of temperature in the ground. Tautochrones are lines connecting observations done at the same time of day. Figure 1, from page 34 of Geiger’s online version (PDF page 60) or page 52 of the Sixth Edition, shows a set of tautochrones.
Figure 1. Tautochrones, from “The Climate Near The Ground”. Numbers on individual lines show the time of day. Vertical axis is depth into the ground, and horizontal axis is temperature.
In my hardcopy version it says regarding this Figure:
“Figure [15] shows the diurnal variations of soil temperature on a clear summer day in the form of tautochrones. These observations by L. Herr were taken on 10 and 11 July 1934 for ten different depths in the ground; the temperature variation with depth shown here is for the odd hours of the day. The tautochrones vary between two extremes, roughly defined by the 15 [3:00 PM] and 5 [5:00 AM] tautochrones. …
During the course of the day, the pattern appears to be complicated by the fact that, in the intervening time. the heat a various depths in the ground may flow in different direction. For example, at 2100 hours, the highest temperature is recorded at a depth of 5 cm. …”
Note that as the temperature wave moves deeper into the ground, a couple of things are happening. First, at deeper levels, the fluctuations are getting smaller and smaller. Second, there is an increasing time lag for the temperature wave to reach greater and greater depths.
Geiger provides the following equation that gives the relative size of the fluctuation at a given depth.
(Equation 1)
where z is the depth in meters, s1 is the size of the fluctuations at the surface, s2 is the (smaller) size of the fluctuations at the given depth “z“, t is the total time to complete one cycle in seconds, and a is the diffusivity of the ground in square metres per second. Diffusivity is a measure of how fast the heat moves in a given substance. Solving Equation 1 for z gives:
(Equation 2)
where log is the natural log to the base e.
OK, so the depth at which the size of the temperature fluctuations drop to some fraction s2/s1 of the initial surface swing is given by that equation. Now, what is the time it takes for the temperature wave to get down to that depth? That is to say, what is the lag in the system at depth z? Geiger gives the equation for that as well, which is
(Equation 3)
where t1 is the lag time for the temperature wave from the surface to reach the depth z. Now, here comes the interesting part. Substituting the value for z from Equation 2 into Equation 3, we get the following
(Equation 4)
There are some very curious and useful things about this result.
First, as I had suspected, the lag is indeed a fixed fraction of the length of the cycle. For example, the lag time for the fluctuations of a temperature wave in the ground to drop to half its initial value is 0.11 of the cycle length. If the temperature cycle is 24 hours, the lag time is 0.11 times 24 hours = 2.6 hours. And if the temperature cycle is 12 months, the lag time is 0.11 times 12 months = 1.4 months. Both of these are quite close to the observed lags in the climate system.
Next, note that both the depth z and the diffusivity of the ground a have cancelled out of the equation. This means it doesn’t matter if the temperature wave is moving in stone or sand, or even in some mixture of layers of the two, the lag time for a given loss of fluctuation is the same. I definitely didn’t expect that.
Next, because there is a direct link between the time lag and the size of the reduction in fluctuations, we can calculate the size of the response if there were no lag. In the case of the climate system, the lag implies a reduction in size of about 50%. This would seem to mean that if there were no lag in the system, the full temperature response would be about twice the response that we currently observe with the lags.
Next, this would also imply that for e.g. a 60-year temperature cycle, the lag in the peaks of the cycle would be on the order of 0.11 * 60 years, which is about 7 years. Now, that would seem to imply that if there were a sudden temperature jump we’d see a long lag, since it is akin to a very long cycle. But there’s an oddity in this, which brings me back to the beach and the sand. The oddity is, it doesn’t matter what the ground is doing a meter down. We’re never in contact with the deeper levels. So if there is a sudden temperature jump, the surface of the ground warms quite quickly—and as the example of the sandy beach shows, it is only the top layer of the ground that concerns us. It is only in cyclical fluctuations, where heat is moving both into and out of the ground, that we see a lag. A steady slow increase, on the other hand, wouldn’t show such a lag. At least, that’s my current thinking …
In any case, that’s what I’ve learned over the weekend. Sadly, it’s Monday, so I’m heading back to pounding nails. My next investigation will be to use the marvelous CERES dataset to get a better grip on this question. I can look for example at the lags in the land versus the ocean, which is likely what is giving the “fat-tailed” response. Note that my analysis above is only valid for solids. The ocean is different in two regards. First, it is free to circulate thermally, allowing it to lose energy faster than the land. Second, it is not heated just at the surface, but down deeper. However, I suspect that these two differences somewhat counteract each other, so overall it is following the same type of path as the land, but with somewhat different parameters. But that’s just a guess at this point.
Finally, I make no overarching claims for this lovely result. I’m still struggling to understand the implications of it myself. As always, I’m just reporting my findings as I come across them.
Man, I do love settled science … there are so many unanswered questions. For example … is it just a coincidence that the time lags in the climate system are about equal to the lag time for the fluctuations to reach half of their original value? I suspect that it is not a coincidence, that it is true for any cyclical system in thermal balance. This is because in thermal equilibrium, the amount of heat coming out of the earth has to equal the amount going in, which I suspect relates to the fluctuations falling to exactly half their initial value … but so far I don’t see a way to demonstrate that.
w.
PS—To return one final time to the sandy beach, my natural habitat, the diffusivity of dry sand is on the order of a = 1.3E-7 m^2 per second, with t = 86400 seconds for the cycle length (one day). Using those variables in Equation 2, we find that the depth z required to get only half the temperature swing of the surface sand is only 4 centimeters, or about an inch and a half …
PPS—And yes, I’m sure that there are folks out there who knew this all along … but I didn’t, which is why I’m discussing it.
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What this speculative discussion fails to take into account is the difference between temperature cycles in a particular place (caused by the Earth’s rotation, its orbit, and seasonal movements of air masses and ocean currents) and the overall energy balance of the planet, i.e. the total amount of heat energy stored by the atmosphere, land and oceans. There are, in fact, regular cycles in the overall energy balance of the planet: they are called Milankovitch cycles, caused by orbital and rotational eccentricities of the Earth. There is no evidence that the heat energy balance of the entire planet also fluctuates on a 60-year cycle, nor is there any known physical mechanism that would cause such a cycle to exist. Otherwise, this discussion is quite interesting.
I’m reminded of partial differential equations (with boundary values) some 25 years ago.
Heat equation comes to mind. Is the above analysis seemingly derived from deductive reasoning simply a special case of the heat equation? I might wager it is…but alas, it’s been too long to take a stand.
Anyone care to help an aging person reminisce?
“This means it doesn’t matter if the temperature wave is moving in stone or sand, or even in some mixture of layers of the two, the lag time for a given loss of fluctuation is the same. I definitely didn’t expect that.”
Isn’t this because this is a MODEL and so only an approximation of reality?
HR says:
June 19, 2012 at 10:02 am
Thanks, HR, good question. No, it’s not because it’s a model, that’s the curious and surprising part. It is inherent in the nature of what’s going on. The key issue is that no matter what physical material the temperature wave is flowing in, it flows out at exactly the same speed that it flows in. As a result, it will take the same fraction of a cycle to get to a thermal wave that is, e.g., half the size of the wave at the surface.
w.
So why are there different rates for diurnal, and annual? That’s the claim over the lags.
Willis,
The results that you quote are simply a solution of the heat equation. If you are willing to really get into the weeds, find a copy of “Partial Differential Equations in Physics” by Arnold Sommerfeld. Section 14 is devoted to “The Problem of the Earth’s Temperature” with the full solution given by equation 14.7 that gives temperature versus depth, x, as an infinite sum of partial waves.
u(x,t) = C_0 + 2*SUM(n=1 -> ∞){ C_n * exp(-q_n x) * cos(2*pi*n*t/T + g_n – q_n*x) }
q_n = sqrt(n*pi/k/T)
T = period of the temperature forcing
k = thermal diffusivity
x = depth
t = time
C_n & g_n depend on the shape of the temperature forcing function, sine wave, square wave, …
“We see that the amplitude C_n of the n-th partial wave is damped exponentially with increasing depth x, and that this damping increases with increasing n. At the same time the phase of the partial wave is retarded increasingly with increasing x and n”.
Note that a daily forcing, T small and q_n large, will penetrate much less than an annual forcing, T large and q_n small.
You can apply the same analysis to the heating and cooling of a pure N2-O2 earth atmosphere, but upside down with the heating occuring at the earth’s surface and with the much smaller thermal diffusivity of the air (assuming no convection, a very big assumption!).
John Marshall says on June 19, 2012 at 3:38 am:
“Thanks for your post. Interesting.
But the earth is never in thermal equilibrium is it. The sun is rising and setting. Night follows day. A cyclic system is never at equilibrium by the very nature of the cycles.”
================
No John, The sun is not rising and setting! – It’s warming the Earth equally, or near so, 24 hours per day, however that does not mean the planet receives energy equally from all directions as is necessary for Blackbody calculations.
It is the Earth that turns around its own axis thus presenting itself to the Sun with a “forever changing” absorption-rate. The Equatorial areas of the Sun and the Earth are ‘roughly’ in line.
I just clicked on: http://www.shadowchaser.demon.co.uk/eclipse/2006/thermochron.gif and all I got was a page displaying: “This site is unavailable because it is too busy.” Well, well, well –Watts up with that?
That’s a good question.
There’s certainly a “lag” from peak solar radiation to peak temperature.
However, there isn’t a lag from “sufficient solar radiation to cause warming” to peak temperature.
I generally think of lags as the time it takes for something to have an effect. For example, the time it takes for a piece of electronics to react to a button press. Or the amount of time it takes to get a doctor to call you back after leaving a message. The lag is the time elapsed between input and result. In the case of solar warming, input is continuous until peak temperature is reached so there is no lag between input and reaction.
Willis you are presenting a classic boundary value problem of the heat equation see
http://thevirtuosi.blogspot.nl/2012/05/how-cold-is-ground-ii.html
buy a text book instead of reinventing the wheel yourself.
try eg Geodynamics by Turcotte and Schubert.
Hans Erren says:
June 19, 2012 at 4:14 pm
Well, I don’t know about “Geodynamics” but the site you specified doesn’t show the derivation I show above. It doesn’t show anything near that derivation. If you want to accuse me of reinventing the wheel, show me the previous wheel. All you’ve shown me are the general equations. You have not pointed to a source where it shows that the diffusivity drops out of the equation so that the solution is applicable to any kind of ground, rock or sand or whatever. Finally, if you can find the derivation I show above, show me where someone uses that derivation in discussing the climate and relates it to the similar lags between monthly and daily temperature swings. Because that’s what I’ve done here. Show me where it has been done before.
More to the point … so what? I mean, suppose you find some obscure text where someone has done all of that … so what? I derived what I derived above on my own, I’m happy with it, and I’m bringing that to a public forum to discuss it. Now you want to turn that into a bad thing and accuse me of “reinventing the wheel” … don’t you have anything positive you could be doing, instead of sniping at people who are actually accomplishing something? Can’t you serve up something other than sour grapes?
Because in any case, I’ve dealt with your kind of nay-sayers before, I saw your kind of nonsense coming, which is why I said above,
OK, fine, Hans, so you’re one of the good folks who knew it all along … who cares? Lead, follow, or get out of the way, your easy jibes about “reinventing the wheel” and “buy a text book” go nowhere. If you understand this all so well, take it the next step, pitch in, lend a hand, move it further along.
w.
Hans, one further thing. When I get the comments like ‘someone has done that before’ or ‘it’s in textbook X’, to me that is merely a reaffirmation that I’m on the right path. I find it to be much more valuable to struggle along and figure things out for myself that it is to have it all spoon-fed to me by some text. I like having what the Zen Buddhists call “beginners mind”, free of textbook preconceptions and the dictates of the consensus. Not that I’m dissing textbooks, you’ll note my paean to Geiger above … but I work outside of them, not through them.
Now, clearly, this is not how most folks operate, but I’m a self-taught scientist. In college I took exactly two semesters of physics 101 and two semesters of chemistry 101 and one semester of calculus. That is the entirety of my scientific education. I’ve had to teach myself everything I know, and at the same time work for a living. Which means that I don’t have the leisure that you might have to go buy a textbook and spend three weeks digging to the bottom of it. So I learn what I can, where I can, and I try to relate it to everything else that I know and make sense of it. I’m the ultimate generalist, both in my work habits and in my scientific investigations. Look at the huge range of subjects that my essays have covered, utilizing my unusual method of investigation. And you know what?
Crazy as my method might seem, it has worked very, very well for me. I have done what I consider to be important science, developed new ideas, and come up with interesting and innovative methods of analysis. Some have been published in the journals, but most of them I’ve subjected to a much more stringent and rigorous test. I have put my ideas out on the very public anvil of the web, handed around the hammers, and invited people to see if they can destroy them …
And sometimes they do destroy my ideas, they find errors large or small, and I suffer a very public smackdown … but that’s the price I’m willing to pay to know that my other ideas, the ones people couldn’t destroy, the ones nobody could find a flaw in, are strong and valid.
And knowing that makes me kinda impervious to guys like yourself who want to belittle what I’ve done, who try to patronize me and tell me to buy a book … I know the value of my ideas, Hans, I’ve tested them in the fire.
w.
Hans Erren, I looked at that site (yours?) and if your cycle/time ‘x’ axis is correct then all of you gradients are leaning to the left (deeper – earlier) and should be leaning to the right (deeper – later). Right? Or am I missing something?
Seems just a set of equations can be dangerous if not applied correctly and leads you to wrong conclusions. The danger is because they came from your beloved text, therefore the equations are correct, You then tend to assume results from those equations are, of course, correct and that can be very wrong.
Seems probably just a flipped sign. Easy to do.
Willis, I think you might have flipped a sign. You took equation two, substituted into three, but two is negative and the ‘z’ in three is negative and seems your 1/2π (0.159) in four ends up positive. If that correct it’s no problem here, I’m human too and have that slip occur regularly. Seems more the s2/s1 should have been s1/s2 to make eq. 2 positive, for it seems do want log(1/2) in the eq. 4.
But now to the substance. I downloaded the Geiger book, have read about a hundred pages and that is a well needed breath of fresh air. I’ll enjoy that one, thanks. I follow you up to equation four, then I lose the logic, the sign doesn’t bother me, the result 0.11 doesn’t bother me, I have tracked that lag here at my home over the last two summers and I was getting ~0.108, basically a match. But, equation three says if the depth ‘z’ is zero then the lag ‘t1’ is also zero. Right? This was Geiger saying the first part of the chapter was theoretical and he ends right after showing the actual empirical tautochron relation where you then expand. What’s you logic going from half gradient magnitude ratio across a depth in the soil to the temporal lag at the surface. I’ve tried but can’t seem to find your logical tie between the two and you don’t go very deep right at that point. Can you clarify?
The reason I ask these picky points is this plays directly into a global energy relationship I have working on (on and off) for a couple of months now, and thanks for the link to Geiger’s book, it holds some estimate numbers I’ve been looking for and it does have included a relation to lags also.
freedomfirst: “Anyone care to help an aging person reminisce?”
Paul Linsay: “The results that you quote are simply a solution of the heat equation.”
Hans Erren: “Willis you are presenting a classic boundary value problem of the heat equation ”
At http://wattsupwiththat.com/2012/06/19/a-demonstration-of-negative-climate-sensitivity/#comment-1013667 I flailed about reinventing what’s set forth better in the link Hans Erren provided.
But it remains to provide a solution to the boundary condition set forth at http://wattsupwiththat.com/2012/06/19/a-demonstration-of-negative-climate-sensitivity/#comment-1014451. If any of you could provide one, that would be great.
Willis: While these equations show how the time lag changes as you go down in a solid surface, I think the more relevant equations for the sort of time lags that you are interested in for the climate system is to start looking instead at box models for the climate system.
For example, a one-box model can be written in the form of a differential equation as:
C*dT/dt = F(t) – lambda*T.
Here, T is the temperature, t is time, C is the heat capacity (the specific heat times mass) of the system, lambda is a constant with 1/lambda equal to the climate sensitivity, and F(t) is the forcing vs time. If you have a periodic sinusoidal forcing of angular frequency omega and amplitude F_0, then F(t) = F_0*cos(omega*t).
These equations can be solved exactly in the long-time limit and the result that one gets is that the time lag varies as a function of omega*tau where the time constant tau = C/lambda. The easiest way to write the solution is that T(t) = the real part of (F_0/lambda)*exp(i*omega*t) / (1 + i*omega*t) where i is the square root of -1.
When omega*tau is much less than 1, the temperature response is roughly in phase with the forcing; when omega*tau is much greater than 1, the temperature response lags the forcing by pi/2, i.e., a quarter of a cycle. And, of course, for in-between values of omega*tau, you get a lag somewhere in between 0 and 1/4 of a cycle.
This result makes sense: If the system has a very small heat capacity, the temperature is basically just proportional to the current value of the forcing. If the system has a very large heat capacity, then the temperature increases when the forcing is above its time-average value and decreases when the forcing is below its time-average value.
With a two (or more)-box model, as is presumably necessary to realistically model the Earth’s climate system, things are more complicated. However, I think it is still true that the time lag will be somewhere between 0 and 1/4 of a cycle.
As a correction to what I wrote above, T is the temperature anomaly, not the absolute temperature.