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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Tim Ball says on June 18, 2012 at 3:24 pm:
“Gradually and finally people start to catch up with what climatology understood, taught and was researching 30 years ago. It speaks to my charge that the IPCC has set climatology back 30 years. —— —-
I am glad Willis is doing this, but it is too little too late. The public have moved on. The only lag left is between final exposure of what the IPCC have done and complete withdrawal of funding.“
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Well said Tim Ball, I would like to hear (read) more from you too!
mike G says onJune 18, 2012 at 2:37 pm:
“@Stephen Fisher Skinner
I think deserts are cool at night because there is very little greenhouse gas (water vapor) to slow the heat transfer to space (CO2 being a trivial greenhouse gas).”
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Pre Arrhenius that was correct thinking.
Thanks for the link to the textbook, I will enjoy reading it if it is as empircal as you say.
But I think you are missing a mechanism related to the atmosphere – I assume that is you point mentioning CERES? – that is not really present for solids.
The “thermal diffusivity” is related to the specific heat capacity, which under quantum theory is related to the number of degrees of freedom, i.e. it is not really a constant
e.g. take a look at things like “pressure broadening”…
eyesonu-
yes, they are perfect for tight places – but not the way to put in 6000 nails per day.
@Mike G says:
June 18, 2012 at 2:37 pm
The cold nights in the desert are due to minimal clouds, not water vapor directly. Direct radiation to space is fairly efficient for cooling without clouds. Net radiation heat transfer via the absorption and radiation of greenhouse gases is only a small portion of the heat transferred from the ground upwards. Convection and direct radiation are larger.
Here in Utah in the summer, we don’t typically hit maximum temperature until late in the day. Sometimes as late as 6:00 pm. I wonder why the lag is almost twice what it should be.
The data I have seen such as:
http://en.wikipedia.org/wiki/File:MODIS_and_AIRS_SST_comp_fig2.i.jpg
seem to show the day to night variation only significantly affects the upper few m of the surface. There would be some overturning to that depth, but that is not the main mixing source compared to wind, and currents, which mixes to sever hundred m.
I have wondered the same thing. In doing thought experiments on the subject, I think the behavior would have considerable hysteresis. For example, as insolation increases the ice is going to melt but there is a lot of it. It takes a while to melt thousands of feet of ice. Melting ice just exposes more underlying ice so while polar insolation is high, so is albedo. Earth probably goes through a phase of increased net cooling because more of the sun’s energy is now falling on ice when the pole wobbles so it points more directly at the sun. And this is exactly what we see. The coldest portion of the ice ages seems to be right before “breakout” from them. I am assuming for this thought experiment that for albedo purposes, it doesn’t matter if you have a mile of ice or a yard, the albedo is the same.
That is until the surface starts to become exposed and at that point things should warm very rapidly. Now, at the same time this is going on, as the ice is melting from the North American continent, the rotational pole is moving slowly Northward. This is effectively moving the edge of North American ice Southward causing it to melt even faster. At the LGM, the rotational pole would have been at about the Northern edge of Ellesmere Island due to mass distribution and wobble might have been a lot different, too. But in any case, once more of North America becomes ice-free and once ocean begins to flood into much of the part of Northern Europe that is now submerged, albedo begins a hasty decline. This causes even faster migration of the rotational pole until finally it is where it is now.
So even though the ice regime might change from accumulation to ablation, it is going to take a long time to melt trough 5,000 feet of ice at Chicago. So while the insolation is increasing, the albedo remains about the same and this causes an increase in net energy reflection into space until that ice is gone and the muddy, rocky, lifeless surface is exposed again which rapidly heats in the sun.
vukcevic says:
June 18, 2012 at 3:52 pm
eyesonu says:
……..
I usually search with google scholar, some time ago came across ‘Advances in Geophysics’ (google books) section ‘Atlantic air-sea interaction’ by J. Bjerknes, lot of interesting stuff in there.
=========================
I agree there is a lot of interesting stuff in that. It stopped at page 70 and I then wanted to begin again at pg 84.
There is a lot on ocean currents and temp / heat if anyone is interested. Understand the Gulf Stream. That was particulary interesting; 3000 ft of warm water. Surprising origin/feed.
Copywrite 1964 when science was just that, science.
Now it’s time for Willis’ recommended book. Long night ahead.
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.
Very interesting, Willis. This brings to mind fractals. As always, I look forward to reading your musings.
Sleepalot says:
June 18, 2012 at 3:24 pm
I keep looking at this temperature graph taken during a solar eclipse.
http://www.shadowchaser.demon.co.uk/eclipse/2006/thermochron.gif
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Wow. 20 C temp drop over ~90 min and then a corresponding increase. I have not seen that graph.
Clouds?
It seems to me that it should be stated as “free to circulate physically” rather than your statement; “free to circulate thermally”. Though it does do that as well.
gnomish says:
June 18, 2012 at 3:02 pm
Thanks, gnomish. I’m not sure of the parallel, as diffusivity is measured in m2 sec-1. It is thermal conductivity divided by thermal capacity. Thermal capacity is the bulk density times the specific heat.
Actually, I use a Hitachi framing gun and Senco finish and roofing guns, plus a cheapo Harbor Freight palm nailer … all air driven, though, I am behind the times I know …
w.
W.
You are using non-linear models. I am aghast!
Conrad
ShrNfr says:
June 18, 2012 at 2:30 pm
Thanks, ShrNfr, I’ve noted it in the head post.
w.
Tim Ball says:
June 18, 2012 at 3:24 pm
Thanks, Tim, always good to hear from you. Regarding your last comment, I follow the old ways there, you know what they used to say,
… or something like that …
w.
“The beach air is hot, and your feet can still get burnt.”
You probably didn’t mean it as it is written but insolation and sand type have much more to do with hot sand than air temperature.
What an outstanding posting! All true science begins not – as too often today – with “I believe”, but with “I observe” followed by “I wonder”. Willis Eschenbach observed that the time-lags of the diurnal and annual temperature cycles were a near-identical fraction of the cycle length and wondered whether they might be connected. It is, as he says, a lovely result, As the Italians say of all glittering hypotheses such as this, “Se non e vero, e ben trovato”.
How I long for the day when science is once again done just as Willis does it – by observing, measuring, counting, wondering, and then applying established theory to the results so as to establish a refinement of the theory, or even a new theory altogether.
Before the Age of Reason, science was too often subordinated to the party line promulgated by the governing class. After it, we are back to the Middle Ages, and the party line is rebranded as a “consensus”.
If we lose the use of reason because we become too terrified of reputational damage to question the party line, we lose the West, and we lose that glorious faculty, unique in the visible creation, that separates us most clearly from our fellow-creatures and unites us most closely to our Creator. Without reason, we are no longer human, and we no longer approach the divine.
For as long as men of learning, of enthusiasm and of curiosity like Willis Eschenbach dare to ask, “I wonder” when others are chanting “I believe”, there is still hope. It is postings like his that make WattsUpWithThat the unmissable phenomenon that it has deservedly become.
the lag between max solar input and max temperatures varies with latitude….for example I am in Trivandrum at about 8 degrees north….over here it is hottest in April when the sun is directly overhead
“The data … seem to show the day to night variation only significantly affects the upper few m of the surface.” – Leonard Weinstein June 18, 2012 at 6:03 pm
About 70% of short wave solar radiation is absorbed in the top metre and can be associated with diurnal warm layers of up to several metre and with diurnal thermoclines . Those features have prompted the definition of a Foundation SST (normally at depths of 5-10m) for dealing with satellite SST errors. But diurnal variation in mixed-layer depth associated depth penetration of short-wave radiation can be up to 50 metre. (typical ref: Yoshimi Kawai et al, Diurnal SST Variation and Its Impact, J Oceanography v63, 721-44, 2007)
gopal panicker says:
June 18, 2012 at 8:44 pm
Thanks, gopal. You are right about the temperature peaking in April, but the solar insolation at 5°N where you are peaks in March, not April …
SOURCE: NASA
w.
[UPDATE: My bad, I wrote 8N which was correct and then showed 5 North. You are 100% right, here’s the actual data:
Lat Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Ann
— — — — — — — — — — — — — —
8 382 409 432 437 429 421 423 430 430 413 387 372 413.8
w.]
Tim Ball says:
June 18, 2012 at 3:24 pm
“……..The only lag left is between final exposure of what the IPCC have done and complete withdrawal of funding.”
Well said Mr Ball. We can only hope that the lag isn’t overly extended. The Global Socialists and Ecofascists have damaged the cause of science, of that there is no doubt…. But it isn’t the first time that science has been corrupted, nor will it be the last. However, as long as there are enquiring minds, good science will be done.
I also saw the reports of the very rapid temperature drops during totality of eclipses. It is apparently a sharp enough chill that eclipse watchers are recommended to bring a jacket even in warm environments.
It also implies a radiation rate for heat loss. In dry climates the temperature climb from the evening low to the peak temp of the day is typically less than 30 deg F (17 deg C), which is comparable to the heat loss shown in only the time duration of the progression into totality.
Where is all that CO2 backradiation when you need it ??? /no so much sarc
Larry
Hi Willis,
http://wattsupwiththat.com/2012/05/20/premonitions-of-the-fall-in-temperature/#comment-991087
Re: Time lags and cycle lengths – I’ve written comments like the following since 2008.
Excerpt::
The ~~4 year cycle in this 1997 paper is associated with a lag of atmospheric CO2 after atmospheric temperature T of ~9 months, and the rate of change dCO2/dt varies ~contemporaneously with T. This CO2 cycle is caused by biological (photosynthesis, etc.) and physical (shallow water dissolution and exsolution) factors.
http://icecap.us/index.php/go/joes-blog/carbon_dioxide_in_not_the_primary_cause_of_global_warming_the_future_can_no/
Then there is the much longer ~~800 year lag of CO2 after T (as measured in ice cores), which I suspect is associated with the upwelling of deep ocean currents. Note that ~800 years ago was the Medieval Warm Period.
It appears that CO2 lags temperature at all measured time scales.
Each temperature cycle has its own CO2 delay, and its own approximate period (cycle time length).
There may also be one or more intermediate cycles between the above two (the late Ernst Beck believed there was), and other shorter cycles.
I think there is ample evidence of a daily localized cycle, driven by photosynthesis..
http://co2.utah.edu/index.php?site=2&id=0&img=30
The evidence suggests that varying atmospheric CO2 is not a cause of climate change, it is an effect.
I further believe that humanmade CO2 emissions are relatively small compared to natural daily, weekly, seasonal and millennial CO2 flux, and are probably insignificant in this huge dynamic system.
No small irony here – if I am correct, both sides of the rancorous “mainstream” global warming debate are wrong. Both sides assume that humanmade CO2 is the primary driver of temperature, and are only arguing about the amount of warming (climate sensitivity to CO2, feedbacks positive or negative, etc.). If I am correct, both sides of the mainstream debate have “put the cart before the horse”. I think Veizer (2005, GSA Today) already understood most of this.
P.S.
~9 months in a ~4-5 year (El Nino?) cycle is closer to ~1/6.
But then, 1/8 of one Wolf-Gleissberg Cycle is ~one Solar Cycle. And didn’t I read something recently that involved a lag of ~one Solar Cycle?
Two brief comments.
1. It is standard practice at synoptic climate stations to measure soil temperature at different depths down to about 1 to 1.5 m.
2. In hydrological modelling the effect of soil heat flux is well known. During spring evaporation rates are lower than calculated from meteorological data, as soils warms up. During autumn the rates are higher as the soil gives up its heat.