Time Lags in the Climate System

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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Leonard Weinstein

I enjoy your thoughts, and I think following through will give some useful results. BTW, the general convention is: Ln is used for natural log (base e), and Log is used for base 10

Interstellar Bill

“Sadly, it’s Monday, so I’m heading back to pounding nails.”
So that’s what you ‘deniers’ do to earn your huge salaries from ‘Big Oil’ —
‘pound nails’? Are said nails being pounded into innocent little ‘Climate Science’?
Also, this inquiry is scientifically useless because there’s no parameterized supercomputer model of all the soil on Earth.

Leonard Weinstein

Most of the visible light spectrum is absorbed within 10 meters (33 feet) of the water’s surface, and almost none penetrates below 150 meters (490 feet) of water depth. Since the warmer water is less dense (and it is warmest at the surface), the mixing is not driven by buoyancy, but only by wave and current action. The vast majority of mixing generally is limited to a couple of hundred m, but a very small amount continues to near between 200 m and 700 m.

burtO

“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, ”
It is not because it “takes time to warm the earth”, it is because “it takes long for heat to leave the earth”.
Heat is stored in rocks, sand and stone, which loose temperature slower than the reduction in incoming energy.
For several hours the earth retains enough heat energy to shift the peak temperature beyond the actual peak influx of energy from the sun.
This is not only true for each day but also for each year, when the heat stored during summer months is enough to shift the temperature maximum beyond the maximum solar influx.

Willis Eschenbach

Leonard Weinstein says:
June 18, 2012 at 1:30 pm

Most of the visible light spectrum is absorbed within 10 meters (33 feet) of the water’s surface, and almost none penetrates below 150 meters (490 feet) of water depth. Since the warmer water is less dense (and it is warmest at the surface), the mixing is not driven by buoyancy, but only by wave and current action. The vast majority of mixing generally is limited to a couple of hundred m, but a very small amount continues to near between 200 m and 700 m.

Thanks, Leonard. Actually, you’ve missed one part of the equation. This is that like the atmosphere, the upper layer of the ocean, the “Mixed Layer”, overturns. The atmosphere overturns during the day, and the ocean overturns at night. This is the main mixing mechanism for the mixed layer, and it is actually quite efficient. As I mentioned above, this is why the temperature constant for the ocean is different from the land …
w.

However, while writing this I just noticed that an older edition is available as a FREE DOWNLOAD!
Whoa — 23meg at 180KB/s…
Okay, it’s worth the time. *right-click-save-as — click!*

Willis Eschenbach

burtO says:
June 18, 2012 at 1:35 pm

“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, ”

It is not because it “takes time to warm the earth”, it is because “it takes long for heat to leave the earth”.

Egads, sirrah, had you not chopped off half a sentence in what you quoted, it would have been obvious that I am talking both about the time to heat the earth, and the time it takes for the heat to come back out of the earth. Here’s my entire untangled quote:

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.

Selective quoting as you have done is not a good thing …
w.

Stephen Wilde

“The ocean is different in two regards. First, it is free to circulate thermally, allowing it to lose energy faster than the land”
Did you mean slower ?
The thermohaline circulation is around 1000 years long.
On the other hand such circulation at the surface allows faster evaporation so perhaps you had that in mind.
Anyway, I think we have multiple lags in the oceans individually around the world and in the global oceans combined.

MangoChutney

Interstellar Bill says: June 18, 2012 at 1:27 pm
“Also, this inquiry is scientifically useless because there’s no parameterized supercomputer model of all the soil on Earth.

Yes, but we took a handful of sand and extrapolated

Willis Eschenbach

Stephen Wilde says:
June 18, 2012 at 1:52 pm

“The ocean is different in two regards. First, it is free to circulate thermally, allowing it to lose energy faster than the land”

Did you mean slower ?

Sorry for my lack of clarity, I was referring to energy gains and losses in the mixed layer. Because it overturns nightly, it can bring warm water to the surface to cool by radiation. This speeds up the energy loss. However, this is counteracted by the larger thermal mass and the depth of the warming. I haven’t done an analysis of that yet, it’s one of the things suggested by this analysis.
w.

Stephen Skinner

“The oddity is, it doesn’t matter what the ground is doing a meter down. We’re never in contact with the deeper levels.”
Isn’t there a relationship with the lower levels when it comes to how quickly or slowly the surface cools, or rather how quickly the surface ceases to heat the atmosphere that it touches? I have noticed how quickly a sand beach cools down while a nearby tarmac road or brick wall is still giving off heat after dark. I’ve thought this is why deserts are cool at night as there is very little stored heat in sand.

To the climate of the Northern Hemisphere an 11 lag between summers Icelandic Low and the AMO is of critical importance, and yet it is known only to few casual readers of the WUWT, while the climate science and the ‘experts’ are blissfully ignorant, the reasons for its existence are even more obscure.
http://www.vukcevic.talktalk.net/theAMO.htm

HankHenry

I’ve always wondered about this. When people talk about the surface temperature of the earth what exactly does that mean. It seems like surface *AIR* temperature is merely a proxy for the true surface temperature of the earth.
Let’s next ask if we need to factor in the temperature of the ocean depths and ocean crust, to get an estimate of the surface temperature of the earth. If you look at trends this might not be important, but whenever I see people doing that elementary zero dimensional model of the earth’s surface, I wonder if they don’t overestimate when they say the average surface temperature of the earth is 12, 13 or 14 degrees C.

eyesonu

Wow, Willis, the man with a hammer nailing the ’cause’ to the wall.
Willis, I enjoy your posts. You combine observations, reasoning, and common sense as well as clear and understandable presentations. These charastics are missing from those pushing the ’cause’.

ShrNfr
mike G

@Stephen 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).

AndyG55

Interstellar Bill says: June 18, 2012 at 1:27 pm
“Also, this inquiry is scientifically useless because there’s no parameterized supercomputer model of all the soil on Earth.”
And just what do you think all that hammering is about ?? Willis has PLANS !!!! 😉

Willis:
The equation for temperature profile into the ground is the “Error Function”, which is the generic solution for the convective boundary condition on the surface of a “semi infinite” solid (conduction) boundary.
Classic combined convection/conduction mathematics.
Max

AndyG55

@Steve Skinner,
Its been several years since I did soil properties at uni, but I remember that these things are related to :
grain size, and grain size distribution
water content
type of grain (if the grains are “sticky” or not)
carbon matter
compactness
etc etc etc
so yes, the type of soil has a huge bearing on the heat holding capacity.

AndyG55

that “carbon matter” should of course read “organic matter”
early morning down here !

gnomish

Another interesting topic. In other fields ‘diffusivity’ is called ‘thermal transfer coefficient’ and is measured in W/(m2K). As you know, temperature is not heat.
I hope you’re not really pounding nails.
I have an extra paslode cordless if you were not joking and are serious about getting work done.

eyesonu

@ vukcevic
Is there a source or reference that I could read to explain the causes of high and/or low pressure systems and why they tend to move or remain stationary? The thought came to mind again while reading link to your post above.
It’s that inquiring minds need to know stuff again.

crosspatch

vukcevic says:
June 18, 2012 at 2:09 pm
To the climate of the Northern Hemisphere an 11 lag between summers Icelandic Low and the AMO is of critical importance

11 of what unit of measure? I think this is years if I remember what I saw of your work in the past. Ok, just confirmed that by visiting the provided link. 11 year lag.

NickB.

Hey Willis!
Nice work as always. It’s been a few years but you might recall an exchange of e-mails from a few years back about a thought experiment (The Concrete Earth). IIRC, you sent me a chart of albedos of various natural surfaces… not sure if it was from the book you mention here.
Anyway, I think I finally found a source for how to model what we were discussing (i.e. surface and subsurface temperature profiles for various natural surfaces vs. manmade surfaces).
Hope all is well – keep up the good work!

Lee

and in the end….the heat you take….. is equal to the heat…..you make

Sleepalot

I keep looking at this temperature graph taken during a solar eclipse.
http://www.shadowchaser.demon.co.uk/eclipse/2006/thermochron.gif

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.
All of the issues discussed here were included in the textbook I was involved with in 1989. 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.
Lags speak to the ongoing problem of climatology – cause and effect. Maybe soon we will hear about the lag between Earth’s peak Gaussian strength and temperature, daily and annually or the many other lags that require investigation and explanation. If you want a good overview you might begin with Oliver and Fairbridge’s Encyclopedia of Climatology published in 1987 and a far better more complete and up to date summary than all the IPCC Science Reports combined.
It is going to take a long time to wash of the stink of politics. If and when we do we will find that we have even less data than 30 years ago because in order to achieve their political science they adjusted, threw out, truncated, or lost, large amounts of data. They closed data collection sites so even if re-established them continuity is virtually impossible.
I am glad Willis is acknowledging the work of Geiger. Let me add to the list the names of Thornthwaite, Penman, Budyko, Herman and Goldberg, Imbrie and Imbrie, Claiborne, Eddy, Douglas, Koppen, Grove, Libby, Labitzke, Van Loon, Emiliani, Bryson among many others and of course, Lamb who must roll in his grave at what happened at the Climatic Research Unit.
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.

eyesonu

gnomish says:
June 18, 2012 at 3:02 pm
I have an extra paslode cordless if you were not joking and are serious about getting work done.
======================
Ever try a pneumatic ‘palm nailer’? Easiest way to drive 16’s or 20’s if you need ease of use and accessibility. Use bulk nails and takes about 3 seconds to sink a 16.
So far I haven’t smashed a finger with one. That really hurts. Smashing a finger, that is.

burtOhoy

“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.”
It takes an 8.hour morning to warm a back yard, and since the sun reduces power at a slow pace after peak power, the influx of energy from the sun will overpower the leaving of thermal energy for a couple of hours into the afternoon barbecue.

David, UK

I feel like I’ve been taken to school – except that school was rarely as interesting as this! And thanks for linking to the PDF, Willis. Times like this a Kindley-type device would be useful!

Bill Illis

The seasonal lags for the ocean are about 82 days, freshwater lakes are about 49 days and land surfaces are 34 days.
You can turn some of these equations into joules/m2/second. At the height of the day, the sun is shining in at something like 960.0 joules/m2/second, but the ground is accumulating that solar energy at an exceedingly tiny number; something like 0.007 joules/m2/second. Even though there is a lag of 2.6 hours, the rate that the energy is accumulating (and then leaves at night when the Sun goes down) is several orders of magnitude smaller. The lags are really very, very small rates of accumulation/net loss (small compared to how much the incoming energy changes over a 24 hour epriod.) It is big part of the picture; energy flows over time, that are not being addressed in the theory.

Philip Mulholland

Willis,
You might be interested in this statement by M. Fourier (1827) in Memoire sur les temperatures du globe terrestre et des espaces planetaires p578

In the parts of the envelope nearest the surface the thermometer rises and falls during each day. These diurnal variations cease to be sensible at a depth of 2 or 3 meters. Below this one can only observe annual variations, which themselves disappear deeper down.
If the rotation of the earth about its axis became incomparably faster, and the same occurred for the movement of the earth about the sun, the diurnal and annual variations would cease to be observed; the points of the surface would acquire and keep the same fixed temperature as the deeps. In general, the depth which must be attained for the variations to cease to be sensible has a very simple relation with the period which drives the same effects at the surface. This depth is exactly proportional to the square root of the period. It is for this reason that the diurnal variations do not penetrate further than a depth 19 times less than that observed for the annual variations

Translated by W M Connolley

Brian H

More coincidence? 1/8 = ½^3, so 3 degrees of freedom/dimensions.
Thx for the D/L link and warning. 13 seconds on my ancient P4, 15 mbps, 518 pages. 😉 Compared to graphics and video files, kinda trivial.

Interesting stuff Willis, the thought going through my mind is is “mama don’t take my Tautochrones away”.
I wonder if the lag is also present in planetary Milankovitch cycles.

Alexander Harvey

Hi Wilis,
If the load was purely conductive the temperature would be in phase with the forcing, i.e. a phase lag of 90º for a sinusoidal signal.
For a purely capacitive load it would be 90º and for a purely diffusive load 45º.
That is as measured at the surface and is true for all frequencies (provided pure means pure)
A spacial mix of mostly conductive and mostly capacitive loads will be somewhere between 0º and 45º but will vary somewhat with frequency.
There is a file abstem3 (?) that contains all the gridded normals (the monthly climatology) for the Hadley temperature dataset. So it contains the average seasonal component so you can see how the phase varies around the world. The annual signal is cleanest at aroung 45 North or South where the semi-annual signal is weakest.
You will see that the phase of the annual cycle varies quite a bit from ~25º in Mongolia (highly land locked in the direction of the prevailing wind) to ~70º near about 3/4 around the equator bound leg of each gyre.
That is my memory serves me well.
Alex

Lawrie Ayres

Out at Whitecliffs in western NSW is an entire community living underground. No artificial warming or heating and very pleasant abodes as well. Shows that humans can adapt to anything and that observation beats theory. We won’t have a few breeding pairs in the Artic after all, just dig. Of course rabbits, moles, wombats and meercats have known all about this for eons.

Alexander Harvey

Oops Wrong Angle!
A spatial mix of mostly conductive and mostly capacitive loads will be somewhere between 0º and 90º but will vary somewhat with frequency.

Alexander Harvey

Not my finest hour:
If the load was purely conductive the temperature would be in phase with the forcing, i.e. a phase lag of 0º for a sinusoidal signal.
I am sure you can sort it out.
Alex

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.

AndyG55

“Lawrie Ayres says:
Out at Whitecliffs in western NSW is an entire community living underground.”
yeah, not somewhere where you want to be outside in the middle of a hot summer day, just lots of plies of heat absorbing and emmitting hard rock and rubble. Near zero moisture. Pretty close to being in an oven.
I went there once, not interested in going again !

Larry Ledwick (hotrod)

Willis, you might also want to do some reading on adobe construction. In a book on adobe construction I read many years ago they mentioned this thermal lag. It is actually central to determining the ideal thickness of the adobe wall to use in a home. If the wall is the right thickness the peak temperature thermal pulse will reach the interior surface of the adobe wall early in the morning when additional heating would be needed, and the cool thermal pulse from the early morning hours will reach the interior surface of the adobe wall during the heat of the day.
At the proper thickness the wall acts as a thermal capacitor charging and discharging with a time constant that is exactly opposite of the exterior temperature cycle.
This is why wine cellars interior temperatures approximate the ground temperature although the walls are only about 2 ft thick, they stabilize the interior temperature at the local annual average temperature.
I was noticing the same thing if you look at the climate records for an area and find the historically hottest day of the year, you will find the local climatic time lag with respect to the summer solstice. The same applies to the historically coldest day of the year with regard to the winter solstice. I am sure that local micro climate and seasonal flow patterns will slightly shift the local climate cycle time lag slightly due to changes in prevailing wind directions and such but I doubt it differs more than a week or so for most locations on the surface of the earth.
As far as thermal lags for water, you might also want to look into studies aimed at solar thermal storage ponds as they have lots of actual test data for how the water pool heats and cools and how that changes with depth. Highest water temperatures are achieved with relatively shallow water with a bottom that can be directly heated by the sunlight as there is less water being heated by the solar isolation in a relatively shallow body of water, than a deep water pool that is heated through out a considerable depth.
Hurricane studies show this hot shallow water can partially compensate for land fall in swampy areas where the abundance of relatively hot water counter balances the normal cut off of heat supply to the hurricane at landfall.
Interesting information!
Larry

Bernie McCune

Willis
I have only skimmed over your material, downloaded this book and skimmed over the book’s table of contents. WHAT AN AMAZING BOOK. As is often the case these old meticulous efforts are really well done. Not to say that I do not dig into your work each time it appears and always take away new ideas and insights. I will be looking at your material again and this book again and again. Thanks for showing us this fantastic resource and of course all your interesting analysis. Nothing more to say right now than thanks again!
Bernie

Bill Parsons

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…

Not since I discovered how islanders and coastal dwellers make free use of unoccupied beach real estate in

So Many People … So Little Rain
Posted on March 10, 2012 by Willis Eschenbach

But that is neither here nor there.

eyesonu

vukcevic says:
June 18, 2012 at 3:52 pm
==============
Thanks, I going there now.

Thanks Willis for this. My dog is smarter than most, he always digs a few inches into the sand where it is cooler before lying down on hot days too.

RiHo08

Willis
I am curious as to why “pi” appears in an equation. I wonder about circles and stuff like that. I don’t understand why a temperature wave needs “pi” to determine a time when temperatures will reach such depth.
The other element is a meter below the surface which to me represents the “frost line” below which I can bury my well’s water pipes and hope they don’t freeze during the winter. Is there something magical about three feet below the surface that represents a lag in temperature penetration? That is, during a normal winter, the temperature to freeze my well’s water pipes will not penetrate to the 3 foot level? Even a very cold surface temperature, just as long as the time lasts only a winter and not all year long?

Thank you Willis for yet another good article. –
“How heat is transmitted through the Earth’s crust” is a complicated study and there is not much written about it since ‘Fourier 1824’. I have mentioned his writings before as I have learnt a lot from it and feel I shall have to re-read him.
However originally I come from a place in the world which is positioned approximately 60 degrees north (60° N) and there, back in the 1940s, the yearly frost that solidified the ground reached down to approximately 2 metres (or meters as you may write it). – Well, that frost was thawed out by late May – early June. Of course it was the water in the soil and not the soil itself that froze solid during the winter.
But, further north, in places where the rays from the Sun have less impact there is something else called “Permafrost” where the temperature of the soil is less than 0° C year after year.
Wikipedia says the permafrost can reach as far down as 1493 meters (m), which is a heck of a depth. Here where I live now (around 50° N there is no permafrost at all and I can only assume it is all down to the Sun.
So a question that bugs me now is: “Does the heat from the Sun percolate down to accumulate at the centre of the Earth, or is it more likely that there is a never-ending energy source down there that keeps the fire burning for billions of years? (Is it maybe burning Uranium or Thorium – no, it surely cannot be fossilfuels?)
I really must re-read Fourier – but – as you have started this line of study I shall be watching the progress with interest.

Greg

“from a time when people actually measured things and thought about them”
And from a time when people understood visual display of quantitative information. What a gorgeous graph.

Doubting Thomas

@Interstellar Bill aid “there’s no parameterized supercomputer model of all the soil on Earth.”
There is also no accurate parametized supercomputer model of the oceans, ionosphere, trophosphere and gulf stream that has sub-routines for calculating the effect of incoming Solar winds/waves.
Which is why most of the Nasa modeling scientists found themselves jobless in 2003.
USGS and various mining departments globally have reasonably geological topographical infrared models which could be used to create a global supercomputer model, however I think the article stipulates that this is an inconvenient truth, therefore ignored by climate change “scientists”,

Leo G

“… the mixing is not driven by buoyancy, but only by wave and current action.”- Leonard Weinstein June 18, 2012 at 1:30 pm
No, the mixing is also driven by thermal instabilities arising from the diurnal variation in heating in the vertical profile- one of the reasons the depth of the well-mixed layer varies with latitude.