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.
I’m having a difficult time conceptualizing the shape of the diurnal temperature cycle. It can’t be sinusoidal right? Sun comes up, reaches a peak, sets…. But At night, radiation drops to zero and remains at zero till morning.
Maybe I’ll go back and take a closer look at the temperature plot posted earlier during an eclipse.
“A steady slow increase, on the other hand, wouldn’t show such a lag. At least, that’s my current thinking …”
You can only define a lag time (or phase lag) if you have a cyclic (essentially sinusoidal) change since this produces a responce of similar forn to the driving force.
The responce to a step change is an exponential. You cannot define a phase or time lag in the same but this is still a lagged response.
It may be interesting to look at the tropical zone as gopal panicker suggests. The equator has 6 month cycle with geometrical maxima at the equinoxes , not 12m as it is outside the tropics, so this may be an interesting test of your 1/8 lag.
Tim Ball says:
June 18, 2012 at 3:24 pm
I used to tell my students a lab experiment would prove insolation doesn’t penetrate far into the ground by quickly pulling up worms and measuring how quickly they reacted by blinking.
Did you also tell them that, if they pronounced “gullible” very slowly, it rhymed with “orange”?
Willis, I have two words for you that might provide answers to your observations – Thermal Conductivity. It involves a time quotient which can be used to explain your observations.
As an engineer, I have been involved in Thermodynamics on a real world level for most of my adult life, and it amazes me that most of these ‘scientists’ involved in climatological studies ignore,dismiss, or attempt to refute the laws of Thermodynamics – a science in its own right which is centered on the study and use of heat. Thermodynamics is, in a large way, related to Climatology; for example it relies on models to predict the effect of temperature differences, heating and cooling, and energy differences. The big difference is that the models in Thermodynamics accurately predict the outcomes while the Climatological models have yet to predict an accurate outcome. One of the big failings of the models in climatology, from my perspective, is that they refuse to look at the problem from a thermodynamic point of view. If you look at the climate as a gigantic and extremely complex heat engine, I believe you will be better able to predict the outcomes of any changes in climate. For example, saying that global warming will cause extreme weather conditions is a fallacy from a thermodynamic perspective because the amount of work you can get out of a heat engine is based on the difference between the ‘hot’ side and the ‘cold’ side. In other words what is the temperature difference across a weather front – that will determine how violent the storms will be. The interesting thing is that for meteorologists the thermodynamic point of view is more valid than the climatological viewpoint. Climatology can produce models which will accurately predict climate change, but until they go through all the experimental challenges that Thermodynamics went through, adopt proper scientific methodology, and take into effect all the extra complexities involved instead of getting dragged into political decisions they rank no more higher than a Cult in my opinion.
That graph of the temperature fluctuation posted by sleepalot during an eclipse is quite revealing. Near the vernal equinox in the middle of the Sahara 12 C temp drop, with the lag quite evident. Pretty cool
Willis said:
I think there can be no difference between a steady slow increase and a cycle of infinite length.
Therefore it can have no impact at all (or after a lag of 0.11 * ∞ years)
“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.”
This result in eqn 4 is not so surprising as it appears to be at first. The diffusivity and depth are what determines the ratio of s and s1 so the latter quantites are sufficient and result from the physical, thermal properties of the medium.
The model is simple so the result is simple. It all assumes a homogenous diffusivity throughout the depth profile of course.
Your observation that this 1/8 cycle lag ( phase lag = pi/4 ) seems to apply at very different scales is notable. Looking at how different SST is from pi/4 may give some indication of how feedbacks make it differ from this passive material model.
Another thought provoking post from Willis. Many thanks.
“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.”
The reason for that phase lag is that even when the driving force has passed its maximum (noon for example) it is still well above the cycle mean and remains above it for a full quarter of a cycle. The max temperature will lie somewhere between the point of max driving force and the point at which if falls to the cycle mean.
Just where in this interval it lies, depends of the restoring forces cooling the system and the intertia (thermal in this case) of the system. Similar arguments show why it’s coldest in mid Feb not 21st December (NH).
Thanks Willis. Reminds me of the time I was based in South West Africa, a few miles north of the Orange River on the edge of the Namib Desert. I became interested in astronomy and used to walk out into the desert at night and scoop a shallow trench to lie in. Much cooler for observing the stars above. I was warned about scorpions but I knew from past experience regular doses of whiskey would keep all insects away.
@HankHenry says:
June 18, 2012 at 2:24 pm
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Further to the observation made by HankHenry
When considering the extent of so called GHG warming (said to be about 33degC), I question the correctness of assessimg ocean temperature solely on the basis of SST and not taking account of the average temperature of the ocean.
After about 4 billion years of heat input from the sun and ocean over turning, the deep ocean is only about 4 degC. This is material. If ocean temperature was more uniform (say if the deep and mid oceans were nearer the range of 12 to 15 degC), the prospects for ice ages would greatly diminish.
The temperature of the deep ocean comes back to bite and permits the development of ice ages.
This is an interesting article and something I’ve been thinking about for some time.
The daily variation tells me two things. First is the lag, and secondly the rate. If we look at the rate of cooling after the sun sets, is quite rapid. I take that as meaning that the earth responds very quickly to changes in the heat input.
Now looking at the lag after the solstice, I’m hard pushed to explain why there should be a lag, when the response to energy inputs must be fast, for the day night variation to be as it is.
Why the difference?
eyesonu says:
June 18, 2012 at 6:27 pm
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And yet Trenberth and the Team would have one believe that DWLWIR is greater than solar irradiance. Funny that all those GHGs didn’t keep and maintain temperatures up for very long.
Since the Earth and oceans started off much hotter, I would have thought you’d realise there must be some other reason for the deep ocean being at 4 degC.
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.
Yeah, I suspect that the heating of the earth from the sun since about 1750 when we came out of LIA, would have a time lag equivalent about the same as the seasons and diurnally, that is about 25% of the heating of the system (about 6 weeks after the summer solstice, about 2-3 hours after noon).
So if the sun increased its output to the earth from ~1750 to ~1950, 25% of 200 years is about 50 years so the temperature might stablise about the year 2000.
Temperature has been stable since 2000.
FYI 4 Willis – diurnal overturn is limited to about the top 1 meter
The Near-surface Layer of the Ocean: Structure, Dynamics And Applications
By Alexander Soloviev, Roger Lukas
http://books.google.com/books?id=4p6gNsSs-vIC&pg=PA250&lpg=PA248&ots=8HlIBxX8UP&dq=ocean+diurnal+overturning
It would be interesting to see the tautochrones for the moon. Apollo 15 & 17 missions placed regolith heat transfer experiments with temp probes at regular intervals between surface and up to 3 meters (or as deep as they could drill). The experiments radioed data back to earth for 4 years. Last I checked some sort of SNAFU at NASA.com had made the experiment data unavailable but it was there a year or two ago and available to the public.
Average weather for Tuktoyaktuk in June:
http://weatherspark.com/averages/28386/6/5/Tuktoyaktuk-Northwest-Territories-Canada
As per the link:
The sun is visible all day, varying from 2° above the horizon at 3:00am to 43° above the horizon at 2:50pm, which is solar noon.
The coolest hours of the day are from 4am to 9am with the coldest at 7am, at which time the temperature is below 3°C three days out of four.
The warmest hours of the day are from 1pm to 8pm with the hottest at 5pm, at which time the temperature is above 3°C three days out of four.
The median cloud cover ranges from partly cloudy (65%) to mostly cloudy (85%).
At clearest time, the 2am of the day, the sky is clear, mostly clear, or partly cloudy 6% of the time, and overcast or mostly cloudy 11% of the time.
At cloudiest time, the 10am of the day, the sky is overcast, mostly cloudy, or partly cloudy 65% of the time, and clear or mostly clear 23% of the time.
The sky is typically clearing in earnest by 9pm.
Nick “Why the difference?” my guess would be something to do with all that water sloshing about in the oceans acting as a massive damper. Home as I call it 🙂
Nick “Why the difference?” my guess would be something to do with all that water sloshing about in the oceans acting as a massive damper. Home as I call it 🙂
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If that is the case, then since the quoted lag was over a month for the annual solar effect, the prediction for the day night variation should be negligable. When the sun goes down, it should take over a month for the effect to be felt. Since it doesn’t there must be another reason for the longer annual lag.
Willis, another very interesting post – hammering the nails is obviously good for the grey matter.
What would the tautochrones look like in the mirror image ie the atmosphere? Does the lapse rate rule supreme? I guess the difference is that the earth/soil/sand remains fairly constant in composure whereas the atmosphere is ever changing. So the atmospheric tautochrones would also be ever changing.
Steve R says:
June 19, 2012 at 12:16 am
That graph of the temperature fluctuation posted by sleepalot during an eclipse is quite revealing. Near the vernal equinox in the middle of the Sahara 12 C temp drop, with the lag quite evident. Pretty cool
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Glad you post this. When I chose to comment on that and made a quick look back at the graph to check the temp drop I mistakenly read (red) the wrong line. It was in fact a 12C temp drop and not a 20C as I stated. Still an impressive drop. Good eyes on you. 😉
Amybody ever measure the back radiation during an eclipse? Should be the same, All the sky is visible. But it is the lack of direct heating that causes any temp drop.
I’m not sure that these should necessarily be called lags.
The sun’s EM energy is strong enough to continue heating the earth’s surface until 3pm during the day and until August (in the northern hemisphere) during the year.
So, while maximum temperatures of the earth’s surface doesn’t occur at the same time as the maximum EM radiation from the sun, I wouldn’t necessarily use the term lag to describe this.
About a year ago, Anthony used the example of a pot on a stove. The pot will heat while the burner is slowly turned up to maximum, and the pot will continue to heat even after the maximum flame is reached and the burner is slowly turned down.
Willis,
As I frequently do after reading an interesting post such as yours, I follow the comments and then reread the original post. Often the comments will lead to a much greater insight into many other and broader depths of knowledge that are related yet vary somewhat from the main points presented in the primary article. That is what makes WUWT the success it has become.
From those of us with a little common sense (and a hammer 😉 we can relate very well with the way you present your rationales and observations. I guess that time lags of the ground heating/cooling is right before our eyes and would relate to common sense. You have called our attention to the obvious. But the greater good is that you have quantified the lapse rates to a degree which focused our attention on the obvious that we took for granted. We see your reasoning as you approach the issue which is in the same way we would do so. It helps/makes us understand. You are the one who shows us the obvious. You do a very good job in that respect and I thank you sincerely.
I downloaded the free edition of Rudolph Geiger’s much-updated 1927 classic, “The Climate Near The Ground” but didn’t stay much on topic as there was a lot of interesting stuff to look at. Science was so different then. It was real.
wobble says:
June 19, 2012 at 7:42 am
“I’m not sure that these should necessarily be called lags.”
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If not lags, just what would you call it?
Peak temperature lagging maximum solar input sounds like a lag to me.
It might be interesting to see if a diurnal signal can be picked out of the ARGO surface data. I tried to pick one out from the Norwegian Sea in June when there’s 24 hours sunlight, but did not succeed. If the ARGO floats make multiple observations for each surfacing then someone might be able to detect a signal using first differences.