Guest Post by Willis Eschenbach

I’ve been thinking about thermal lags in the climate system. Everyone is familiar with thermal lag in everyday life. When you put a cast-iron pan on the stove flame, it doesn’t heat up instantly. Instead, the warming process starts instantly, but it takes some amount of time to actually up to a stable heat. And when you turn the heat off, it doesn’t cool down immediately. That’s thermal lag.

Now, Dr. Shaviv claimed that I didn’t understand harmonic analysis and the lag between the sun and what it is heating. I’ll return to that question further down. For now, thermal lag is generally modeled as an “exponential” lag. In an exponential lag, say when you take a hot pan off of the flame, it doesn’t become cold immediately. At first it cools fast. Then as time goes on and it nears room temperature, it cools slower and slower. That “fast at first, slower later” form is characteristic of an exponential process.

In the calculation of an exponential lag, the delay in the response is governed by a time constant called “tau”. The larger the time constant tau, the greater the lag time, and the slower the pan will either heat up or cool down.

With that as prologue, here’s a graph of the heat being turned on and then off under the pan on the stove (black/yellow line), and the temperature response of the pan itself (lines colored blue to red).

Figure 1. Example graph showing how a pan heats up and cools down. Time is in minutes. The fire is shown by the black/yellow line. The fire goes on under the pan at about time t = 8 minutes, and comes off again five minutes later at about t = 13 minutes or so. The time constant “tau” is measured in whatever the underlying time unit is, in this case minutes. Amplitude in arbitrary units.

Figure 1 shows graphically the change in pan temperature, using different colored lines for different values of the time constant tau. The blue lines show small values of tau. A small value for tau would be the situation with say a small thin aluminum frypan, which would rapidly come up to heat and stay at temperature until the fire is turned off. At that point, it cools down quite rapidly. We say that it has small “thermal mass”, so it heats up quickly.

At the other end of the spectrum, the red line shows a large time constant tau of 9 minutes. It would be characteristic of say a big, heavy cast iron cook pot. It would have lots of thermal mass. In the example above, note that the iron pot heats up so slowly that it doesn’t even get up to heat before the fire is turned off.

This illustrates at least three points about this kind of exponential thermal behavior—exponential action, changing shape, and changing size.

First, the action happens exponentially. This means quickly at the start, and then slowing down as time goes on. For example, a hot pan taken off the fire cools rapidly to begin with, but as it approaches room temperature, the cooling slows.

Second, and this is an important point. In general, the shape of the exponential lagged response (the pan temperature) is NOT the same as the shape of the impulse (the on/off of the fire). In the red lines in Figure 1 above, you can see how the square-wave on/off thermal impulse of the flame (black/yellow line) is transformed into a kind of shark-fin shaped thermal response in a heavy cast-iron cookpot.

Third, if the time of the impulse is short enough, the amplitude of the response is smaller than that of the impulse. In other words, in Figure 1 the blue line gets as high as the black/yellow line, but the red line doesn’t get that high before the time comes when it starts dropping again.

There is a special case, however, where the exponentially lagged response has the same shape as the impulse. This is when the impulse has a sinusoidal (sine wave) shape. An example of this would be the solar strength as the earth moves nearer to and further from the sun over the course of the year. Over the course of a year, solar strength varies sinusoidally about 23 W/m2 on a 24/7 global average. It peaks in early January, and bottoms out in early July.

Anyhow, I got to thinking once again about the effect of exponential lag on a sinusoidal input. So I made up a graph like Figure 1, except using a sinusoidal input instead of a square wave input.

Figure 2. Lagged versions of a sinusoidal input. The input is the black/yellow line in the back of the drawing, which represents a flame under the pot of varying strength. The lagged versions showing temperature variations in the pan are shown in color. Time units are quarter cycles. If the full cycle is one year, tau is in months.

Now, I was very happy when I first saw that result. I was elated because I’d never been able to calculate the thermal lag directly from the observations. Oh, I could fit a lagged curve to the results, but to me that’s unsatisfying. And in this graph, I noticed a very curious thing that I’d not known before.

This is that the peaks of the lagged versions of the input wave fall exactly on the original sinusoidal impulse line (black/yellow). This was important to me, because I’d never been able to figure out how much smaller the lagged response wave was than the size of the original impulse. For example, the peak in the earth’s temperature lags behind the peak in the sun. This makes the thermal response smaller … but how much smaller? The graph gave me hope that I could calculate it, and after much investigation of the various questions, much walking in the forest and lots of headscratching, I’ve figured out a heuristic formula that relates all three variables—the time constant tau, the observable lag in the timing of the peak temperature versus the peak of the input, and the scale factor reflecting the loss of amplitude due to the lag. I’m sure some better mathematician than I can provide the actual functions, but the ones I have are very accurate. I’ve appended the functions at the end.

And there is a lot more to learn from Figure 2.

One thing is that the longer the lag, the smaller the resulting thermal response. Expected, but good to see just how it works.

Another is that the resulting lagged signal is also a sine wave. In this special case, the exponential lag doesn’t change the underlying sinusoidal shape of the input impulse (black/yellow line). An increasing time constant tau just pushes the sine wave further and further back in time after the impulse, and scales that sinusoidal impulse ever smaller with increasing lag.

This has an important corollary. If we are looking for the result of sinusoidally varying forcing, the thermal response will have the same shape as the input forcing, but it will occur later in time.

An important observation is that with increasing tau (blue to red, tau of 1 to 9), the colored lines get closer and closer together. As a result, no matter how big the time constant tau gets, the observable lag will never be greater than a quarter of a cycle. For a yearly varying input like the solar energy, that means that the peak temperatures in the ocean and the land can never lag more than three months behind the peak solar … and if they did they’d have zero amplitude. Figure 3 shows the lags for values of the time constant from one to nine.

Figure 3. Closeup view of the upper left section of Figure 2. Vertical black lines show the time lags in the peak of the observable response with values of tau varying from one to nine. Time units are quarter cycles. If the full cycle is one year, tau is in months.

Notice how the vertical lines bunch up at the right. No matter how large tau gets, the actual lag in the response peak is always less than a quarter cycle.

So now I have have developed the tools do the lag math. I can take a look at the lag between solar input and thermal response, and from that I should be able tell you the time constant tau, as well as the scale factor lambda, which relates the solar input to the change in temperature. Let me put this to a bit of practical use.

I looked for starters at the Northern Hemisphere ocean temperatures, and the corresponding NH sun. I’ll use the data from the CERES satellites. The data is monthly. To increase the accuracy of the calculation of the observable lag, I first fit a spline to both monthly datasets. I spline them at 360 points per year, then I average the points. I do this for both the solar data, and for the ocean temperatures. Then I graph them against each other as follows:

Figure 4. Scatterplot of the monthly average northern hemisphere solar anomaly versus the monthly average of the northern hemisphere ocean anomaly . Data has been splined before averaging with a resolution of 360 points per year.

So here’s how the procedure works. Using the splined version shown in Figure 4 allows us to measure the lag between the solar forcing, which has extremes in December and June, and the lagging ocean temperatures with peaks a couple months later, in February and August. This is about two months. Splining it to 360 points puts the measurements in degrees. There is a 61° lag in the summer, and 58 degrees in the winter. So the lag is almost exactly sixty degrees.

Then, using the relationships established above, I find the following:

Observed lag angle: 61° summer, 58° winter

Time constant tau: 3.3 months

Tau-dependent lag scale factor (amplitude reduction from no-lag condition); 0.48

Thermal sensitivity lambda factor (accounting for lag scale factor); 0.067°C per W/m2

I have not mentioned the “lambda” factor in the lagging equations. This is the transfer function or value that converts from the range of solar energy input swings to the range of the resulting temperature swings. Lambda in this case is .067°C per watt per square metre (°C per W/m2). This means that when the solar energy changes by 100 W/m2, the ocean changes temperature by about 7°. Note that this thermal sensitivity (7 degrees per 100 W/m2) is about twice that shown in Figure 4 above. This is because of the lag scale factor—the lagging alone cuts the response about in half, which means that the actual sensitivity is about twice what the observations show.

Now, having calculated those values, I used them to do my test of whether my conversion functions were correct. These were my functions to convert between observed lag, tau, and lag scale factor. Holding my breath and hoping I’d squashed all of the bugs, here’s what I got:

Figure 5. A comparison of the observed NH ocean temperatures (black lines with circles) with the lagged and scaled solar energy input (red lines).

As you can see, it appears that my method is quite accurate. The values for the time constant tau and the lag scaling factor were calculated directly from the one observation, that of the observed lag between the peaks. It was done without using a fitting process of any kind. This indicates that my functions are, well, functioning well …

Next, let me compare the thermal response of the northern hemisphere land with that of the northern hemisphere ocean.

Figure 6. As in Figure 5, but also including the land temperature response. Note the different vertical scale from the previous figure.

Both the land and the ocean get the same sunlight. As you would expect, the land temperature varies much more than the ocean temperature. Here are the calculated thermal lag parameters for the land, including ocean comparisons.

Observed lag angle: 25° summer, 22° winter (ocean 61° summer, 58° winter)

Time constant tau: 1.0 months (ocean 3.3 months)

Lag scale factor (lagged amplitude divided by no-lag amplitude); 0.91   (ocean 0.48)

Thermal sensitivity factor lambda (accounting for lag scale factor); 0.137 (ocean 0.067°C per W/m2)

Why is the thermal sensitivity factor lambda higher for the land than for the ocean? The answer is, they vary in the thermal mass which is affected by the yearly swings. Curiously, this can be calculated directly from lambda, the thermal transfer function. In a twelve-month dataset, where tau is measured in months, the mass involved in the swings is the mass of a depth of ocean in metres which is 8 times tau. In other words, on land tau = 1 so an amount of mass equal to about 7 metres of ocean is involved in the annual swings. In the ocean, tau = 3.3, meaning the water down to 3.3 * 8 = 26 metres is involved in the annual swings.

Now, I speculated a while back that the lag at the daily level is related to the lag at the annual level. I just realized that I can now answer this question by simply examining a graph of the hourly surface temperature where I live. If my speculation is correct, the time constant for the individual station data should be on the order of that for the entire land mass. That was a tau of 1.0, which is 0.52 radians. I wouldn’t expect too great a fit, because one is a point measurement and the other is a hemispheric average. Anyhow, here’s the Santa Rosa data.

Figure 7. Hourly surface temperature, Santa Rosa, California.

Now, using the tools I’ve developed I can take just the information visible in this graph, no digitizing, and calculate the time constant tau for that situation. Tau in Santa Rosa for the daily data turns out to be 0.48 radians, compared with 0.52 radians for the monthly values for the entire NH land …

So it does appear that the time constant (in radians) is period-invariant, meaning that it is related to the lag angle and it doesn’t vary with the length of the cycle.

GENERAL CONCLUSIONS

It is possible to use solely the observed lag between an impulse and the thermally lagged reaction to directly calculate the time constant tau as well as the scale factor resulting from the lagging alone. I’m sure there are mathematicians out there who both knew that and can derive the formulas mathematically.

Knowing tau and the lag scale factor allows us to calculate the thermal transfer factor lambda.

• In exponential decay the time constant tau is also known as the “e-folding time”. This is similar to the concept of “half-life”. Half-life is the time it takes for something to decay to 0.5 (half) of the original value. Similarly, e-folding time is the time it takes for something to decay to 0.37 (1/e) of the original value.

Let’s see what that means in practice. Over the land, the time constant tau is about one month. That means that after a month, the original signal is down to 0.37 of the original value. So far so good.

So where is the signal after two months? Well, another month has passed, so it is now 0.37 of its value at the start of the month, which is 0.37 times 0.37 = 0.37^2. And so on down the line. This means that it is dropping off fast. In general it falls off as e^(-t/tau). In practice, this means that with a time factor tau which is 2 * pi / 3 or less (a tau of 4 in a twelve-month cycle), any signal will decay to within a percent or two of zero within one cycle.

CLIMATE CONCLUSIONS

Since I’m looking at observational data for the temperatures, the data contains all possible responses to the rhythmic solar forcings. These include but aren’t limited to resonances, reactions, changes in clouds, feedbacks, slow and fast exponential increase and decay from past inputs, regime shifts, variations in water vapor and thus the overall greenhouse effect, the speed of the throughput of the energy, and all of the rest.

Now, what are we measuring with the thermal transfer coefficient lambda? We are measuring what the thermal response of the system would be if there were no thermal lag. But if there were no thermal lag … then it must be an equilibrium thermal transfer coefficient, or quite close to one. This is interesting because this implies that the lambda values for the no-thermal-lag condition must be close to what is called the “equilibrium climate sensitivity”. This is usually expressed as the temperature variation arising from a forcing change of 3.7 W/m2, as this is the calculated change in forcing from a doubling of CO2. For the ocean this gives us a “climate sensitivity” of .07 °C per W/m2 * 3.7 W/m2 = 0.26 degrees per doubling. The land figures are 0.14 °C per W/m2 * 3.7 W/m2 = 0.5 degrees from the forcing change from doubling.

Now, I see this as a measure of the strength of the climate thermal regulation system. The average swing in incoming solar energy in the Northern Hemisphere is two hundred watts per square metre peak to peak. The average swing (weighted ocean/land average) in no-thermal-lag NH temperature is 7°C/.48 lag factor * .7 + 26°C/ .9 lag factor * .3 = 19°C peak-peak. On the other hand,  If we figure the blackbody temperature change from that 200 W/m2, if there were no thermal lag that would swing the northern hemisphere’s temperature by 37°C peak to peak … and instead it only swings about half of that, about 19°C. This is from a combination of fixed and variable ground albedo, variable cloud albedo, emergent thermoregulatory phenomena, biosphere variations, and all of the other ways that the climate responds to the changes in forcing. And it’s a good thing we do.

Finally, is there a long-term residual that keeps going for a long time? Possible, to be sure. For example, there could be 11-year cycles from sunspots, or longer cycles from the swings in solar activity. Dr. Shaviv is looking for such longer cycles in the satellite sea level data, and I wish him well. It’s a crazy world out there, and I’ve been surprised before at what people discover. However, given the small size of the thermal time constants, in the range of one to four, it seems such signals would die out for practical purposes in fairly short order

My best regards to everyone, 2:17 am of a foggy night. On the radio, Roger Miller playing through the headphones, “I’m a man of means, by no means … king of the road” … it’s that kind of night for me, and I hope for you as well.

[UPDATE] the always reliable Joe Born writes in the comments:

<blockquote>When you’re talking about a one-pole (“one-box”) system, as Mr. Eschenbach implicitly is (the climate systems isn’t one, but for his purposes we’re assuming it approximates one), the attenuation theoretically is 1/sqrt(1+(tau * omega)^2), and the lag is theoretically arctan(omega * tau)</blockquote>

I’ve used these to re-write my formulas. My old heuristic formula gave the tau of the ocean as 3.27 months … the actual formulas from Joe give a value of 3.30 for the tau of the ocean. So my conclusions are still solid.

I replied to Joe:

Joe, here’re my new formulas based on your equations. They all take native units (e.g. hours, months) for the lag (peak forcing to peak response), tau, and the period. So a function call of lag2tau(2,12) means 2 months lag in a 12-month year, with the result being tau in months.

Attenuation is a decimal from 0 to 1. The empty function calls call the function with the default values, it’s how I test my functions.

tau2lag=function(tau=3.3,period=12) atan(tau*2*pi/period)/(2*pi/period)

tau2lag()

[1] 1.998003

lag2tau=function(lag=2,period=12) period*tan((2*lag*pi)/period)/(2*pi)

lag2tau()

[1] 3.307973

tau2atten=function(tau=3.3,period=12) 1/sqrt(1+(tau*2*pi/period)^2)

tau2atten()

[1] 0.5009052

lag2atten=function(lag=2,period=12) tau2atten(lag2tau(lag,period),period)

lag2atten()

[1] 0.5

tauconvert=function(tau=3.3,period=12,newperiod=24) tau/period*newperiod

tauconvert()

[1] 6.6

Note that omega is equal to 2*pi/period, again with the period in native units. Also, since tau is expressed in units of omega it is not conserved when you go between say a 12-month and a 24-hour situation. Use the “tauconvert” function to convert tau between periods. I’ve deleted the old R functions at the end.

w.

My Perennial Request: If you disagree with someone, please have the courtesy to quote the exact words you disagree with. This allows everyone to understand exactly what you object to.

Math Notes: The main equation I use calculates the change in temperature per unit time as

where the subscript shows time, T is temperature, F is forcing, tau is the time constant, and lambda is the transfer function or coefficient.

Adopting the terminology that for any variable X, the variable ∆X is defined as:

And making the substitution of alpha for the term involving e

the main equation simplifies to

The first term on the right hand side, involving the forcing change ∆F, is the direct action of the forcing. “Lambda” is the transfer function, which scales the input to the output. The second term on the right, involving the temperature change ∆T, is the lagged action. Incoming forcing is partitioned between the two branches (immediate action and delayed exponential decay action) by the alpha and (1-alpha) terms. As tau gets larger, the immediate direct action decreases and the length of the lagged action increases.

If tau is zero, alpha is zero, and there is no lag. The response is simply the input times lambda, the transfer factor that relates the range of the thermal response to the range of the input forcing.

Tau and lambda are the only parameters in Equation 1.  My goal in this investigation was to be able to calculate tau and lambda directly from nothing more than the delay between impulse and response. In that, I’ve been successful.

Dr. Shaviv’s Explanation

I said above that I’d return to Dr. Shaviv’s objections to my previous postings, which I append below:

Eschenbach has no clue on what the meaning of harmonic analysis is. He still thinks I assumed the sea level should be in sync with the forcing…

Suppose the solar forcing is Delta F = F0 exp (i 2 pi t/P)

The heat in the oceans (per unit area) will be (assuming feedback is small)

Q = int Delta F dt = – i P F0 / (2 pi) exp (i 2 pi t / P)

The sea level, if it is only affected by thermal expansion, will then be

Delta h = alpha Q = – i alpha P F0/(2pi) exp (i 2 pi t / P)

thus,

Delta h = alpha Q = – i alpha P/(2pi) Delta F

It seams like an algebraic relation, but in fact, the -i gives a 90° phase shift which describes the integral(!) If one take the real part of the above equation (to get the physical quantities), one finds

Delta F = F0 cos (2 pi t/P)

Delta h = alpha P F0/(2 pi) sin (2 pi t / P ) = alpha P F0/(2 pi) cos (2 pi t / P – pi/2 )

That is, the integral appears in a harmonic analysis as a phase lag of 90°. In real life, this angle is not exactly 90° because of additional processes taking place, which was the whole point of the analysis!

So again, Eschenbach’s analysis, compared the sea level to the solar forcing and found there is a poor fit. It is because the sea level is supposed to lag by 90° after the solar forcing! Eschenbach still does not acknowledge this.

Also, Eschenbach’s claim that I compared to the sea level and not the rate ignores the fact that a harmonic analysis is expected to give a 90° lag if the sea level is the integral of the forcing, and this is exactly the case.

SOURCE

OK. I’ve corrected a couple of Dr. Shaviv’s math typos, in red. I hate math typos, easiest thing to do, and I certainly hope I don’t have any in my math above.

There are two problems in the above explanation. First, Dr. Shaviv is discussing two kinds of lags, without noting the distinction. Second, except for in the math, he uses “forcing” to mean “CHANGE in forcing”.

I’ll start with the question of the two lags, and I’ll use his terminology from his math. The first lag is the 90° lag in a sinusoidal system between the change in forcing Delta F (∆F) and the thermal response Q. The second lag is the thermal lag discussed above. The first sinusoidal lag is a result of the curious fact that the integral of a cosine is a sine, and a cosine is just a sine shifted by 90° … a couple facts that I’ve known since I taught myself calculus in high school.

Note that this lag he is discussing is between ∆F, the change in forcing, and Q, the result. However, in a sinusoidal system the same lag exists between ∆F, the change in forcing, and F, the forcing itself. In other words, although there is a 90° lag between ∆F and Q, there is the same delay between ∆F and F. As a result, there is no lag at all between F and Q.

Now, recall that Dr. Shaviv said above that:

So again, Eschenbach’s analysis compared the sea level to the solar forcing and found there is a poor fit. It is because the sea level is supposed to lag by 90° after the solar forcing!

Here you see the second problem. Dr. Shaviv is incorrectly using the term “forcing” F to mean the CHANGE in forcing ∆F, the quantity which in his equations above he calls “Delta F”. Yes, sea level is supposed to lag Delta F by 90°. But if there were no thermal lag, there would indeed be no lag between sea level and forcing F.

This is a continuing problem with Dr. Shaviv’s claims, which I’ve pointed to before. For example, he also says above:

… a harmonic analysis is expected to give a 90° lag if the sea level is the integral of the forcing, and this is exactly the case.

Not true in the slightest. Look at his own math. In his analysis above, he doesn’t use the integral of the forcing (which would be “int F dt”). His exact statement was:

Q = int Delta F dt

He says that the change in sea level is alpha Q, and he says this sea level change 90° out of sync with the solar forcing F.

But take another look at his equation immediately above … what is the integral of Delta F? … I didn’t see it the first time through myself, but as you might expect the integral of the derivative of the forcing F, which is his int Delta F dt above, is the forcing F itself. That means that Q = F, and it means that in the absence of thermal lags, the sea level changes and the forcing indeed are in sync.

So when Dr. Shaviv’s says that “sea level is the integral of the forcing”, he is simply wrong. His own math shows that sea level is NOT the integral of the forcing F. It is the integral of the change in forcing Delta F ∆F.

The other kind of lag is the lag I discuss above, the lag between the peak forcing F and the peak thermal response. If there is no thermal mass then the thermal response is immediate and the lag is zero. As shown above, the thermal lag can never be more than a quarter cycle, and the thermal lag is a function of a time constant tau.

With those two kinds of lags in mind, consider Dr. Shaviv’s statement:

That is, the integral appears in a harmonic analysis as a phase lag of 90°. In real life, this angle is not exactly 90° because of additional processes taking place, which was the whole point of the analysis!

The phase lag between cos(x) and int (cos(x) is 90°. Always. Exactly. In real life, it is still exactly 90°. It is not changed by “additional processes taking place”. It is a mathematical fact. However, it is not much considered, because in real life the forcings and the responses are NOT 90° out of phase as Dr. Shaviv claims. They are generally in phase, with the response slightly lagging the forcing.  If the sun goes stronger and weaker, and the sea level is a function of the sun, we expect the change in sea level to be somewhat thermally lagged behind the forcing … not 90° out of phase plus or minus something.

Let me be clear that I think that Dr. Shaviv knows the difference between the forcing F and the change in forcing ∆F. Otherwise, he couldn’t do the math. I don’t know why he keeps confusing them over and over in his responses.

Finally, I was amused by a commenter on Dr. Shaviv’s quoted passage who said:

I’d rather [Willis] took some math classes. His previous teacher was not up to the task. (He identifies as self-taught). He’s just a few semesters away from a “who knew” moment.

It’s true, I’m self-taught in math. When I was a junior in high school, I finished all the math courses that the school offered. So my math teacher, Mr. Hedgji, bless him, gave me a calculus textbook and said “Go for it”. He helped me when I had questions, and otherwise left me alone to go for it.

So I did, and I never stopped.

I notice that the anonymous commenter couldn’t find any actual math errors of mine to comment on, so he resorted to personal judgements. Meanwhile, I’m the one who puts my math work out here on a regular basis so that folks like the charming commenter can snipe at it, and so that serious mathematicians can point out my flaws. And I’m the one discussing the problems with Dr. Shaviv’s math. Not only that, but I give both the commenter and Dr. Shaviv full opportunity to find any problems with my work and point them out in public. Strange as it may seem, I welcome people finding errors in my work. Folks like Mosher and Nick Stokes and Dan Hughes and too many other commenters to name have saved me literally years of wasted effort running blind alleys by pointing out logical, mathematical, theoretical, or other errors in what I was doing, and my hat’s off to them for their due diligence. It’s no fun at all to have my errors pointed out in public … but it’s far better than the alternative.

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LB
August 24, 2015 2:12 am

For once you can experiment with the climate. What you need to do in order to work out the lag in the system is turn the heating on and off, and watch how the temperature changes in response.
So how do rig up the earth to do the experiment?
Ah yes. At night the heating is turn off, and in the morning its turned on again.
So we can see from the diurnal temperature change how rapidly the atmosphere reacts to changes in any inputs.
That varies from 4C in high humidity tropical areas to 57C in desert areas.
So the lag is very small

Mike
August 24, 2015 10:34 am

All you need to do to get the exponentially lagged response is do a convolution with a suitable exponential decay. Convolution can be easily done, it’s just a kind of weighted running average. In this case the weights are the exponential. You make it as long as is needed to for the values to be too small to worry about.
This is all explained in full detail here where the method was applied to the volacanic forcing of Mt Pinatubo.
https://climategrog.wordpress.com/2015/01/17/on-determination-of-tropical-feedbacks/
This has been posted at Judith Curry’s site and linked here several times.

george e. smith
August 25, 2015 1:21 pm

Well Willis’s computation seems to be a steady state response to a steady state drive signal, by a one pole low pass filter, with a 3 dB cutoff frequency of 1/(2pi.n.tau). where n is Willis’s 1-9.
So no need to do a time domain analysis. But then the question is, why would one assume that the low pass filter is one pole with a single time constant.
But as far as “lags” are concerned, there really isn’t any lag.
Everything happens almost instantaneously, but then there a whole host of things that happen sequentially, and the apparent “lag” is simply all the instants of time involved in handing off the baton to the next process. Each molecule has to bang into its nearest neighbor and so on, and since they have a finite free path velocity, it takes some time to travel the gap to the next molecule to bang into. But each collision is a very short event; all of which can be calculated either by QM or maybe even classical dynamics.
But Mother Gaia is not sitting around asking; “I wonder what value of n.tau Willis is using today ??
That is the whole point with heat. Everything is chaotic and disorganized, but each action, reaction and interaction is happening as fast as it possibly can. Only the limit value of (c) is putting a lid on how fast things can happen.
Now if you really want to do the time response of a transient event, try to feed the single pulse v = Vsin (theta) for (theta = zero to 2pi) into Willis’s single pole n.tau filter.
g

Mike
August 24, 2015 10:36 am

BTW the response does not get “smaller” it just gets more spread out as tau increases.
The area under the graph is always identical.

August 24, 2015 2:09 pm

So, I’ve been doing this with surface data for a handful of years.
This is the day to day change in temp for >23N to <66N lat
Each year is a single plot line.
If you calculate the slope of the lines as they cross zero, you get this.
Next, if you measure the grounds(concrete) temp with an IR thermometer, grass, asphalt, as well as the sky and air temp you can see something like this
Sky is the coldest, then grass, then concrete, finally asphalt, this is the heat of the day, to the following morning. The sloped line on the warmer IR plots is the concrete cooling as a shadow moves down it's length, the distance from the front porch down the walk to the driveway which is asphalt, which is on the south side of the house so it has Sun till later in the afternoon.
Finally Willis's local temp plot?w=720
In you notice late in the night (early morning) the rate of cooling drops off a lot, I can explain that as well.
Notice how rel humidity spikes every night? once it get in the 80-90% range the rate of cooling drops off, because while the ground is closer to air temp, the sky is still really cold.
When you look at a longer time frame, you can see the effect of rel humidity maxing out at night.
Willis, consider a pan warmed from above with an IR lamp, but with a really cold surface behind the lamp for the pan to cool to when the lamp is off, and not measuring just the pan, but the air over the pan.

August 24, 2015 2:10 pm

The first 2 plots are duplicates, Oops!

August 24, 2015 2:19 pm

here’s another interesting graph, this is the difference between yesterday’s raising temp, and last nights falling temp.
I was interested in figuring if there had been any loss of cooling at night due to an increase in Co2. First, on a daily clear sky basis, it cools all night until the sun comes up, then if you compare a complete year for many surface stations looking at whether over all that station was warming or cooling. On average it’s cooled more than warmed for 1940-2014 (almost 70 million samples), 50 of the last 74 years are cooling, 30 of the last 34 years cooled.
Also, the warm years (absolute temps) are the years it’s cooled more than warmed. Co2 could not cause this.

Bloke down the pub
August 24, 2015 2:24 am

Above my pay-grade, but I do hope that by remaining civil we can all learn from this discussion.

Scott
Reply to  Bloke down the pub
August 24, 2015 5:22 am

Actually, the civil discourse. between Willis and Dr. Shaviv indeed shows how lively debate in science is supposed to happen. Gentlemen, please keep it coming!
With time, you’ll solve the problems of the Universe!…

RERT
August 24, 2015 2:25 am

Willis – great stuff as ever!
It seems to me I can take the 0.067 degrees per W/m^2, multiply by the average peak -trough swing in daily average solar forcing, from memory about 0.3, to give an average trough to peak amplitude of the effect of the solar cycle on ocean temperature of circa 0.022 Degrees. That would be the raw number before reduction due to lags, so perhaps 0.01 degrees.
Quite small to pick out of the noise!
Is that calculation legitimate?
R.

Jon
August 24, 2015 2:38 am

Now that’s the scientific spirit in action. Other people’s views and critiques aren’t a threat but a help.

Scott
August 24, 2015 5:23 am

+100

August 24, 2015 4:42 pm

Yea, thank you Willis. Your generosity of time and effort will advance our knowledge and understanding. You are an example of what “scientist” is supposed to mean. But man, your math leaves me awed. Wish now I had actually become educated.

August 24, 2015 2:47 am

RE this angle is not exactly 90° because of additional processes taking place, this might be the case if the “additional processes” influenced the period of a sinusoidal forcing, simply because the lagged value would be sometimes be slightly out of phase which the current forcing, but surely using a simple “harmonic component” rather than the actual solar cycle eliminates period variations as a possible source of phase change?

August 24, 2015 2:55 am

I am reminded of a Feynman comment (from memory) that the most exciting words in science is not “Eureka!” but “That’s funny!”
It’s fascinating to watch the evolution of this investigation, even if all your answers urn out o be wrong. Thanks or sharing.
“Science is a game in which the object is to discover the rules.” -Sylvester Wolfe

August 24, 2015 2:58 am

typo fix:
…even if all your answers turn out to be wrong. Thanks for sharing.

leafwalker
August 24, 2015 7:29 am

Usually attributed to Isaac Asimov:
http://quoteinvestigator.com/2015/03/02/eureka-funny/

August 24, 2015 12:32 pm

The older I get, the more I miss-remember. But old jokes are funny again.

August 24, 2015 4:04 am

Willis,
Here’s a simple electrical analogue – resistor R and capacitor C in parallel to earth, forced with a sinusoidal current I. The voltage (T) across the combined impedance is IR/(1+jωCR), where ω is freq in radians/sec, and j is sqrt(-1). The time constant is RC, and the voltage phase shift is atan(ωRC) – this gives the shift in terms of time constant RC. It’s 0 at low freq, and varies up to 90° with rising frequency, as you say.

Peter Sable
August 24, 2015 3:05 pm

I’ve seen a motorcycle exhaust system modeled in an analog electrical simulator (SPICE). The math is all the same.

Editor
August 24, 2015 4:07 am

Hi Willis. Thanks again for another wonderful post. You may want to add a note about your use of the word anomaly. Many of the readers will understand you’ve taken, for example, sea surface temperature data in absolute form and then zeroed it to highlight the seasonal component, but some readers might think it was derived from sea surface temperature anomaly data.
Cheers.

Phil B
August 24, 2015 4:07 am

The magnitude of the increase in temperature on land compared to the ocean goes a long way toward explaining why the terrestrial modelers are so keen on projecting their land based (adjusted) measurements out over the ocean.

August 24, 2015 4:51 am

Regarding: “For the ocean this gives us a “climate sensitivity” of .07 °C per W/m2 * 3.7 W/m2 = 0.26 degrees per doubling. The land figures are 0.14 °C per W/m2 * 3.7 W/m2 = 0.5 degrees from the forcing change from doubling.”
This ocean figure is true when the variation is rapid enough (full cycle in one year) for largely only the top few or several meters of ocean to warm up and cool down. The top few or several meters of ocean are having their temperature swings being limited not only by thermal mass, but also by mixing with the ocean underneath. If a thermal forcing is sustained over a longer period of time, so that deeper levels of the ocean can have a corresponding temperature change, then the ocean surface temperature change would be greater. Same for land, whose temperature changes are somewhat limited by exchanging heat with the ocean through the atmosphere.
This would predict temperature change in response to a sudden forcing being non-exponential. The presented data point of tau in radians (for land) being about the same for a daily cycle and an annual one indicates that the time constant (in time units) is not constant, but varies with the duration of a cycle or a pulse of thermal forcing. (The presented data point being the Santa Rosa one, re: “Tau in Santa Rosa for the daily data turns out to be 0.48 radians, compared with 0.52 radians for the monthly values for the entire NH land …
So it does appear that the time constant (in radians) is period-invariant, meaning that it is related to the lag angle and it doesn’t vary with the length of the cycle.”)

Peter Sable
August 24, 2015 3:11 pm

You can have a tau that’s orders of magnitude different (e.g. the deep oceans) and not see it in on daily, monthly, or yearly scales. Try applying a huge tau to the periodic or short pulse examples above. You’ll get a flat line. Change the period of the sine wave or length of pulse to 70 years and you’ll see it.
The interesting question is how many taus are in the Earth’s regulatory system.
I’ll also note we simply don’t have enough data to measure the very large taus of the Earth’s regulatory system.
Peter

Frederik Michiels
August 24, 2015 4:53 am

willis if you crank out this with the lag that’s coupled to the “devries/suess cyclic change of the sun, i think you got a winner….
When you would find the way how the earth as a full mass does behave tpo these cycles of the sun (of which the maunder, wolf and Dalton minima were absolute dips and former optima absolute tops spreaded over a 200 year near sine wave and the 1400 year sine wave of the Bond events i think the actual warming can be explaiend by entirely natural forcing….
something to look into?

Ryan
August 24, 2015 4:56 am

We can see the same thing you pointed out on a yearly basis happen on a daily basis. The peak of the sun is somewhere around 1pm but the highest temperature is somewhere from 3-4pm. The coldest time of the day is about 7-8 am after the sun just starts to rise.
Willis, I would like you to do another calculation on top of this study. Please calculate the thermal expansion of the ocean water in the 26 meters that is affected by the up and down in temperature. How much cm or mm can the water rise and fall over the year.

Bill Illis
August 24, 2015 5:09 am

The lags tell you a great deal about what is really going on in the energy system contained in the Earth’s molecules.
The Ocean surface lags behind the solar forcing by about 82 days. In the northern hemisphere, solar energy peaks on June 21 but most of the northern hemisphere ocean surface temperature peaks on September 12.
The Land surface lags behind the annual solar cycle by about 32 days. The lowest solar forcing is December 21, but the lowest Land temperature is January 22.
At 45N, the average mean temperature varies from 20C on July 22 to -10C on January 22. This is equivalent to the land gaining/losing about 0.8 W/m2 per day or about 0.00001 joules/m2/second over the annual cycle.
In the daily cycle. the land air temperature lags behind the Sun by about 3.5 hours. Solar peak at noon, air temperature peak at 3:30 pm.
If one thinks about how much solar energy is coming in at noon on June 21, 960 W/m2 or 960 joules/m2/second (including the average cloudiness), but the actual air temperature is only increasing by about 0.008 joules/m2/second at noon. Almost all of the solar energy, 99.9992% of it, is not going into heating the air, it is immediately being re-emitted back to space and/or is being absorbed into the soil and rock molecules in the surface just as fast as it coming in.
At night time, when there is no solar energy coming in at all, but the air temperature only cools off as if 0.006 joules/m2/second is being emitted out to space. 959.992 joules emitted/lost at noon but only 000.006 joules emitted/lost at night.
On a per second basis, there is a lot going on in the real physical world that is not being captured in climate science. Energy accumulates on a per second basis by extremely minute amounts. The solar cycle only varies by +/- 0.4 W/m2. That is already an extremely small amount so take a extremely small amount and accumulate energy from that variation at even more extreme minute amounts. Over many decades, it could make a difference, but not for a single solar cycle or just a few. It would take 30 or 40 years.

Greg Degi
August 24, 2015 5:23 am

As usual I enjoyed your post. You might consider going to the Electrical Engineering section of your local University’s bookstore and look into “signal processing” and “control systems” texts. A lot of what you have been writing about already has a solid math foundation in those areas.
Greg

Bruce Ploetz
August 24, 2015 5:38 am

Willis, your discussion describes a system that has only one energy storage element. This is very familiar territory to the electronics field, where it would be described as a first-order system. In this case the energy storage element is the mass that is being heated. You could model it as an energy source that feeds a resistor with the output in parallel with a capacitor. (Voltage source+ in series with resistor in series with capacitor returning back to source-, measure the voltage across the capacitor while applying a step to the source). If you can find a free version of Spice you can draw similar graphs and play with the part values – you will see the same relationships you describe.
Of course in the real world there are many energy storage elements besides the mass of the land or sea, and a true working model equivalent in electronic terms would be much more complex. But, since the simple model explains and clarifies the part of the system that is under discussion it is perfectly appropriate to neglect the underlying complexity and focus on the simple broad-stroke relationships. We do this all the time in electronics, deriving a simple relationship that explains the broad strokes then digging into the details when necessary.

beng135
August 24, 2015 10:00 am

Of course in the real world there are many energy storage elements besides the mass of the land or sea

Seems to me practically all the heat storage is in the land & ocean. Other heat storage mechanisms would be pretty much negligible — except massive ice-sheets during the glacials.

Mike
August 24, 2015 1:03 pm

I would say that at least we should consider more than a single slab ocean model, which what the simple linear model effectively is.
If the deep ocean heat sink can be considered as effectively infinite and of constant temperature and the heat transfer from the mixed layer to deep ocean is typified by eddy diffusion proportional to the temperature gradient, this also simplifies to a linear model with slightly modified feedback.
This was discussed by Paul_K and myself on Willis’ last thread.

beng135
August 25, 2015 5:51 am

Mike says:

I would say that at least we should consider more than a single slab ocean model, which what the simple linear model effectively is.

Mike, I agree and I wasn’t ignoring the deep oceans. But I’ve pondered the deep ocean below the mixing zone for some time & it has strange characteristics — temperature well below the avg earth temp. Either vast amounts of cold, sinking water are constantly replenishing all of it (unreasonable given the huge volume of the cold water), or it is remarkably thermally insulated (stratification) from the mixing layer above.

August 24, 2015 5:41 am

Market gyrations today being what they are, I won’t have time to write anything but a drive-by, but I’m sure others will make the point anyway.
When you’re talking about a one-pole (“one-box”) system, as Mr. Eschenbach implicitly is (the climate systems isn’t one, but for his purposes we’re assuming it approximates one), the attenuation theoretically is $\frac{1}{\sqrt{1+(\omega\tau)^2}}$, and the lag is theoretically $-\arctan(\omega\tau)$.
Some time ago Mr. Eschenbach wrote about “tautochrones,” and there he observed how different from the response of the lumped-parameter system he uses here the responses of distributed-parameter systems such as heat transfer through the soil are.

August 25, 2015 6:25 am

Glad to help. (I like to think we also serve who only do the sums.)
I wish I could take a little more time with the post, because I’m not sure I correctly understood “Tau in Santa Rosa for the daily data turns out to be 0.48 radians, compared with 0.52 radians for the monthly values for the entire NH land …
“So it does appear that the time constant (in radians) is period-invariant, meaning that it is related to the lag angle and it doesn’t vary with the length of the cycle.”
If I did, my initial impression is that the behavior sounds distinctly non-one-box-like; I would have thought that one-box-system lag at around the 50-radian-lag point would be much more sensitive to a two-and-a-half-decade frequency range–and that lag would increase with frequency, not decrease with it.
As I say, though, I probably didn’t understand that excerpt, and, if I did, I may not yet be analyzing it correctly. Otherwise, I would wonder how much we should base inferences on assuming a one-box approximation.

toorightmate
August 24, 2015 5:56 am

This dumb old ox is trying to learn more about the variables associated with climate and I get a cooking lesson!

Phil.
August 24, 2015 5:57 am

This is that the peaks of the lagged versions of the input wave fall exactly on the original sinusoidal impulse line (black/yellow). This was important to me, because I’d never been able to figure out how much smaller the lagged response wave was than the size of the original impulse.
This is a classic result and is frequently made use of in correcting for response times of measurement devices (e.g. thermocouples). One way is to simultaneously measure using sensors with different response times, the peaks all lie of the raw data curve which can then be reconstructed.
The reason for this can be easily seen if you look at the differential form of the response equation:
Tg = 𝜏.dTw/dt + Tw
Tg is wire temperature, Tg is gas temperature
At the peak of the measured curve (Tw) dTw/dt=0, therefore Tg = Tw.
If the response time, 𝜏, is known the input curve can be calculated from the measured curve using the above equation.

Mike Jowsey
August 24, 2015 6:05 am

As the bloke down the pub said, above my paygrade, but I got the jist and failed the math. Sounds interesting WIllis. Keep wandering around the forest scratching your head. Much appreciated.

August 24, 2015 6:12 am

I would like to see what results when you analyze hourly dew point data at death valley and La Jolla. The transfer of energy through evaporation/condensation is a big factor in the resulting surface temperature,and should be considered in a lag.

Leonard Lane
August 24, 2015 9:51 am

Good point I believe.

August 24, 2015 8:34 pm

I have found that as Rel humidity gets ~>80% or so, the rate of cooling slows, you can see this as the change in cooling rate in Willis’s last graph, and I have a couple graphs near the top of the thread.

David Thompson
August 24, 2015 6:19 am

It gets interesting when there is more than one lag in the system and even more difficult if you throw in some pure delay. Think heat being carried by ocean currents. Throw in feedbacks and negative feedback becomes positive feedback when the phase shift gets to 180 deg. The response goes from exponential to sine wave. The fact that the oceans have oscillations may indicate 2 or more lags in the system. It’s not an indication of anything being unstable in engineering terms unless the oscillations are growing, closed loop gain at the resonant frequency less than one.
Ltspice is free, good, and has a graphical schematic tool.

Earl Wood
August 24, 2015 6:28 am

Willis,
In Figure 5, there are years where the actual data for the ocean temperature dips lower than or stays higher than the model you developed. Is this the effect of La Nina, El Nino?

richard verney
August 24, 2015 6:48 am

Whilst it is interesting to look at the thermal lag in the system in the short term, as I see matters for those that support the view that solar is a major driver in long term climate, one needs to look at the Holocene.
What caused the Holocene Optimum, the Minoan Warm Period, the Roman Warm Period, The Medieval Warm Period, the LIA?
That is what has to be answered. For example, was it:
Planetary procession?
Changes in CO2 (or other so called ghgs)?
Changes in atmospheric aerosols due to volcanos and/or asteroid impact?
Changes in patterns of cloudiness, and if so what drove these changes (was it the sun or perhaps the earth’s magnetic field)?
Changes in the amount of water vapour, and if so what drove those changes?
Was it the sun?
If not one or more of the above, what could it be?
There is a reason why the warmists have been so desperate to flatten the shaft of the hockey stick and to do away with these variations in temperature/climate over the past 12,000 years.
Planetary procession does not explain the periodic warming events, nor does CO2 if it has been essentially stable (as warmists assert) over the last 5,000 to 8,000 years. Volcano eruptions are mere speculation since there is strong evidence of only a few volcano eruptions over this period, and volcanos seem to have only a short term impact on temperature.
Presently, we may not know the answer, and may be we never will know the answer, but most reasonable people can appreciate that solar insolation received by the surface is a strong bet, when considering the possible presently known/contemplated/foreseen candidates. We just do not know the mechanism but with further evidence and understanding we might see the light.

gbaikie
August 24, 2015 11:41 am

–What caused the Holocene Optimum, the Minoan Warm Period, the Roman Warm Period, The Medieval Warm Period, the LIA?
That is what has to be answered. For example, was it:
Planetary procession?
Changes in CO2 (or other so called ghgs)?
Changes in atmospheric aerosols due to volcanos and/or asteroid impact?
Changes in patterns of cloudiness, and if so what drove these changes (was it the sun or perhaps the earth’s magnetic field)?
Changes in the amount of water vapour, and if so what drove those changes?
Was it the sun?
If not one or more of the above, what could it be? —
Ocean mixing
Ocean surface transparency.
If one could just stir the ocean, it has the largest effect upon global average temperature.
Mixing the ocean causes immediate cooling and long term warming [and rise of sea level due to thermal expansion].
I not going to go into how it’s mixed or how much it is mixed, but just saying if one could thoroughly
mix the ocean, it has huge effect of upon average temperatures. So I think mixing ocean belongs
on the list.
And ocean transparency can effect short term warming which results in longer term cooling- if less
transparent at the surface.
Volcanic eruption can effect this and any fertilization has biological component.
Or just because open oceans currently are very clear and sterile, doesn’t mean this is the constant state of things over the centuries.
But anyhow, what caused the cooling periods appears mostly unresolved.

richard verney
August 24, 2015 4:36 pm

Obviously it has been cooling ever since the highs of the Holocene Optimum. Since the Optimum, the cooling has been interrupted by a number of warm periods, each one peaking at a slightly lower high, ie., the Holocene Optimum was warmer than the Minoan Warm Period, which in turn was warmer than the Roman Warm Period, which in turn was warmer than the Medieval Warm Period, which in turn was warmer than the late 20th Century warm Period (although we do not know whether the latest warm period has peaked).
Some people ‘believe’ (such as Willis – see his article on radiating the oceans which relies upon circular reasoning for much of what is proposed) that DWLWIR heats the oceans, and the energy absorbed in the top few microns is effectively mixed by the action of waves, swell and ocean overturning. But if that is so, why are the deep oceans so cold? Why is the average temperature of the oceans less than 4 degC?
After all, there has been about 4 billion years of solar plus DWLWIR going into the oceans and if there is effective mixing of the energy being imparted into the oceans, one would expect after all this time for the oceans to be far warmer at least over the top 500 or so metres.
I suspect that when we have better understanding, the only energy in the oceans will be seen to be solar (ignoring the small addition of geothermal), and that DWLWIR does not effectively heat the oceans.
I am of the firm view that it is the oceans that drive the climate but they are in turn driven by the sun and the amount of solar insolation received over time.
During the Holocene, there has been no significant movement of the continental shelves and land masses such that the pattern of oceanic currents (and the general manner in which they distribute energy imparted in the tropical seas pole wards) is likely to be similar to that seen today. Whether the rate of these currents (including the thermohaline circulation) has changed over time is obviously not known, nut if it has, then the most likely candidate for that change is changes in the receipt of solar insolation.
Of course, one might expect there to have been changes in the rate of the thermohaline circulation as ice melts and glaciers recede during the Holocene, but that would be a response not a driver.

gbaikie
August 24, 2015 5:48 pm

–richard verney
August 24, 2015 at 4:36 pm
Obviously it has been cooling ever since the highs of the Holocene Optimum. Since the Optimum, the cooling has been interrupted by a number of warm periods, each one peaking at a slightly lower high, ie., the Holocene Optimum was warmer than the Minoan Warm Period, which in turn was warmer than the Roman Warm Period, which in turn was warmer than the Medieval Warm Period, which in turn was warmer than the late 20th Century warm Period (although we do not know whether the latest warm period has peaked).
Some people ‘believe’ (such as Willis – see his article on radiating the oceans which relies upon circular reasoning for much of what is proposed) that DWLWIR heats the oceans, and the energy absorbed in the top few microns is effectively mixed by the action of waves, swell and ocean overturning. But if that is so, why are the deep oceans so cold? Why is the average temperature of the oceans less than 4 degC? —
I don’t think DWLWIR warms the ocean, but were it to do this, any significant wattage should cause evaporation rather than warming. Or DWLWIR is quite different then SW, which transparent to water.
Or DWLWIR would heat to a depth of about 1 mm, whereas SW warms to meters depth- such differences should be easily measurable/noticeable.
As for cold ocean it’s caused by colder polar temperature and polar sea ice.
As for the slow cooling from the Holocene Optimum, generally it’s due to the orbital variation, I imagine.
How exactly that works, I don’t know. There the issue that in some sense we have yet to recover from the Little Ice Age, perhaps none of the warming periods fully recover. Or the Holocene Optimum is a more complete recovery [from glacial period] and latter recoveries not as complete.
But it seems like good bet that ocean circulation/patterns/mixtures are related to it.

mrmethane
August 24, 2015 6:53 am

I guess the work of the much-maligned Dr. Evans will eventually be acknowledged and cited as being relevant. “Lag” and “delay” seem to be somewhat similar in nature.

August 24, 2015 6:57 am

Reminds me of the circuit theory classes taken over 40 years ago. For a linear system, the response to a stimulus is the convolution of the Dirac Delta Function response and the stimulus. Response to a sine wave input will be shifted by about 90 degrees if the inverse of the natural frequency, 1/(2xPixF) is substantially shorter than the time constant of the impulse response. Phase lag will be 45 degrees when the inverse of the natural frequency is equal to the impulse response time constant ASSUMING the impulse response is a simple exponential decay.
Conducive heat transfer happens by diffusion, where the time for heat to diffuse past a surface is proportional to the square of the distance.

Mike
August 24, 2015 12:57 pm

Yeah, that’s it. convolution.with the impulse response. Not forgetting to reverse the convolution kernel in this case since it is asymmetric. For FIR filters the kernel is symmetric and reversal is ignored. ie the kernel is a rising exp not a decaying one.
https://climategrog.wordpress.com/2013/03/03/scripts/
This script will do the exp convollution that gives the linear feedback type response.

August 24, 2015 7:01 am

The comparison between ocean and air responses to energy changes shows how a similar issue arises between moist and wet air, and the problem of averaging temperatures while ignoring humidity. And this is the proper interpretation of Arctic amplification, which runs ahead of the lagged ocean and moist air.
https://rclutz.wordpress.com/2015/08/24/arctic-amplification/

eyesonu
August 24, 2015 8:08 am

An excellent post by Willis.

August 24, 2015 8:13 am

Willis: I detect a chance that you are getting close to the atmospheric/ocean lead/lag to any OVERALL increase or decrease in the energy balance of the atmosphere and the oceans. Ergo, with the 18 year “hiatus” on the land temps and the really, “dead level” Argo data for 12 years, what do you think? Do we have an indirect way of saying “something balances”. (Might be your wonderful thunderstorms! Or natural heat pumps.)

August 24, 2015 8:17 am

OT, latest from ESA
One of the most significant glacier calving events on record.
Satellite images show that the fastest moving glacier in the world shed a chunk of ice measuring around 12.5 sq km this week
http://www.esa.int/Our_Activities/Observing_the_Earth/Copernicus/Sentinel-1/Chasing_ice

Latitude
August 24, 2015 9:10 am

The Jakobshavn Climate Scam
The terminus of the glacier is almost identical to two years ago.
https://stevengoddard.wordpress.com/2015/08/22/the-jakobshavn-climate-scam/

August 24, 2015 8:37 am

Another set of data which in the scheme of things is meaningless.
The expected results post 2000 to present as far as the climate /solar connection is concerned is the climate due to solar activity should have been stable at the high end of the temperature range for the past 30 years or so which is exactly what has taken place.

prjindigo
August 24, 2015 8:50 am

except
When you light a fire it heats the air immediately, that heated air rises and spreads out into the world. Each and every input of heat into the system is IMMEDIATELY processed, some a little more slowly and convoluted than others.
What you’re describing is a ‘firewall’ state, where heat does not transmit through a material with exceptionally high specific heat for a duration of time. Like Space Shuttle bricks or a 3/4 inch quartz drywall barrier.
The Earth’s atmosphere isn’t made of 3/4 inch quartz drywall. The oceans are not space shuttle bricks.
You want to know how much heat is in the Earth’s atmosphere? Ask NASA the total altitude of the Earth’s atmosphere on any given day and that will tell you.
We do NOT live in a closed system (1 gallon gas can) and sea level temperatures are regulated by gravitational force. The law of thermal-dynamics will tell you the rest.
I’ve said it hundreds of times to thousands of people: Temperature at sea level is regulated by GRAVITY. Gravity is the constraining force of the environment of thermal-dynamics of the Earth’s atmosphere.
If your global climate change model isn’t gravity regulated, it will not work.

Paul Coppin
August 24, 2015 9:40 am

When you light a fire it heats the air immediately, that heated air rises and spreads out into the world. Each and every input of heat into the system is IMMEDIATELY processed, some a little more slowly and convoluted than others.

Except, no. Lag may be infinitesimally small, but it is never zero. The example lacks boundaries. Heat energy distributes evenly and instantaneously only in a theoretical model. Fundamentally, this is the crux of our issue with understanding the mechanics of climate. There are SO MANY confounding variables and periodicities, the system is chaotic. Looking for a simplistic cause and effect for global warming, or the lack of it, is a mug’s game, and why the CO2/GHG causation is doom to fail, but for politics.
As a biologist, I watch physicists emasculate their intellect trying to describe climate in simple terms, knowing what the bio and geosphere tells me: aeons of variability, much of it with complex, only loosely stable periodicities, point to a homeostasis that is only so in human chronologies. On a species-specific level, predictablity is attainable, or nearly so. On a planetary level, we’re not even close. Not. Even. Close.

August 24, 2015 7:14 pm

But for all relevant purposes gravity is a constant . It may explain the difference between our surface temperature and top of atmosphere temperature but has nothing to do with to with the ~ 0.3% variation that this fraud is built on .
Brilliant article , the sort of thing which should be covered in undergraduate climate science curricula .
The small lag between temperature and insolation versus claims of decades long lags between increased CO2 and temperature was one of the first things to get thru to me the seriousness of this determined silliness . In 1984 for one of my first annual CoSy gatherings , http://cosy.com/views/cg84invt.htm , I made a plot showing the small lag between a sine approximation to insolation and local temperature near Rochester NY . My interest was as much creating the functions to create the bit-mapped graphs on the 500k APL 68000 Sage computer as it was the subject of the graph . But it has always struck me how little analysis I see of the much greater effect of GHG’s on our variance in temperature as opposed to the mean .
http://cosy.com/views/cg84.gif

Les
August 24, 2015 8:52 am

Just a quick question re Figure 4…. If you take a horizontal line across the plot at 0 degrees, would the area under the line be exactly equal to the area above the line OR would a difference in area reflect longer term trends in warming or cooling?

Les
August 24, 2015 8:53 am

Sorry, that quick question was directed to Willis

August 24, 2015 8:55 am

Once the low average solar average value parameters are met and the duration comes into play in the context of sub-solar activity in general post 2005 then we will see the climate response to solar activity.
I expect this time if /when the low average value solar parameters are met the climate will start to cool.
The item to remember is when the climate changes it changes in a step fashion superimposed perhaps upon the slow gradual trend which becomes established.
The best recent example is the period 1275 ad to the early years of the following century.
All this talk trying to prove or not prove some existence that means anything of the so called 11 year sunspot cycle /climate connection is not where the focus should be namely because there is no solar/climate connection of any significance when the sun is in that kind of operating mode.
Willis we agree the 11 year sunspot cycle and the climate correlation is meaningless.
Willis says below which is for the particular time period and rhythmic solar forcings in contrast to the point I keep trying to make is what happens during prolonged non rhythmic solar forcing periods and why does the climate change so abruptly so often? This is what matters.
Since I’m looking at observational data for the temperatures, the data contains all possible responses to the rhythmic solar forcings

August 24, 2015 9:02 am

Willis assumes apparently that thresholds are not present in the non linear , chaotic ,random climate system.
Willis always picks data to prove his points at a time when the climate system was very stable which is why these kind of studies do not do much for me.
Try applying it to the times when the climate is not in such a stable mode.

Alan Robertson
August 24, 2015 11:41 am

.When might that be?

beng135
August 24, 2015 9:26 am

Really interesting, Willis.
Haven’t groked this enough yet, but seems some pretty basic & crucial climate characteristics can be determined from this technique.

Tom Crozier
August 24, 2015 10:39 am

This explains why a watched pot never boils, option pricing in the financial markets, and a few other things not so easily measured.
It’s interesting that thermodynamic laws can seemingly be used to model human behavior, but I often wonder whether the widespread use of the models in the soft sciences creates short term self fulfilling prophecies.

Mark from the Midwest
August 24, 2015 12:26 pm

Human behaviors, in the aggregate, can often follow patterns that are similar to many of the functions in the natural sciences. But that happens only when the population is broadly defined, (such as all men ages 25-49). It’s really just a function of the nature of the variance across persons, and a poor approach to try to understand processes like consumer choice, etc., etc., etc…

lgl
August 24, 2015 10:42 am

“the sea level changes and the forcing indeed are in sync.”
So when Dr. Shaviv’s says that “sea level is the integral of the forcing” he is right. The integral of the sea level change is the sea level. dH/dt=Q, where H is heat content and Q is energy added. (and Q is the integral of F)

Mike
August 24, 2015 11:03 am

I didn’t see it the first time through myself, but as you might expect the integral of the derivative of the forcing F, which is his int Delta F dt above, is the forcing F itself.

Slow down Willis you’re getting it all mixed up and saying Shaviv does not know what he’s doing again.
deltaF is NOT d/dt(F) , it’s like an ‘anomaly’ , it is not the derivative. Most of your comments based on that and how Dr S does not know his maths are incorrect. I’m not going to quote every one and say what it should be since it all comes from the same mistake.
The resulting change in temperature will lag the change in solar as cos lags sine. I suggest you read the following, I’m sure you’ll learn how to find the lagged response to solar or any other input you like with a simple weighted average:
https://climategrog.wordpress.com/2015/01/17/on-determination-of-tropical-feedbacks/
You may like to look back to the discussions between Paul_K and myself later in comments on your last post.

Mike
August 24, 2015 9:45 pm

∆T = λ ∆F
ie the temperature anomaly is proportional to the anomaly in the forcing.
So ∆F is not the rate of change of F and its integral is not equal to F.

It’s all very confusing.

It does appear that you are confused.

Peter
August 24, 2015 11:15 am

Hi Willis,
I got recently one idea which I would like to share with you. I was surprised by your self regulating tropics theory. I was thinking how it can go with reality of ice ages and more quasi stable states of Earth. I was writing into some discussions about your findings that 26C looks like stable maximum temperature achievable on Earth with current condition of gravity and mass of atmosphere. Then I realized. Gravity is probably not changing, but mass of atmosphere…. So I started to dig. Nitrogen is probably not changing, CO2, Argon trace gases, so remains only Oxygen. It is very hard to find ratio of Oxygen for last 800,000 years. Like nobody cares. I only found this:
http://www.skepticalscience.com/images/CO2_vs_oxygen.gif
It looks like we are loosing Oxygen. In 15 years we gained 30ppm of CO2, but lost almost 300ppm of Oxygen!
I also found this:
http://wattsupwiththat.files.wordpress.com/2012/06/clip_image001.png
It looks like atmosphere is thinning.
Oxygen is lost I think only through burning, burning of Carbon so CO2. As CO2 level is not increasing proportionally to Oxygen that means that natural processes of sequestrating CO2 are working very well. And it can not be photosynthesis, because it would return O2 back to atmosphere. And then I realized: I was writing here few comments about marine life sequestrating Carbon in shells and removing it from biosphere for million years. Basically choking biosphere to death by lack of CO2. We are solely dependent on volcanism returning CO2 back to atmosphere. What I realized recently is that Calcium Carbonate – CaCO3 is sequestrating one atom of Calcium, one atom of Carbon but three atoms of Oxygen… So correction, marine life is able to choke land life to death by lack of CO2 and O2…
I checked content of Ca in oceans is around 400ppm, that means 0.56x10E18kg of Ca, and in atmosphere we have 1.03x10E18kg of Oxygen. That means there is plenty of Calcium in oceans to store all atmospheric Oxygen as CaCO3, just keep adding Carbon.
And guess what Calcium Carbonate creating reaction is exothermic, heating water. I think everybody is now thinking about El Nino. Burning 300ppm of atmospheric O2 in ocean creating CaCO3 should be detectable on ocean’s temperature and heat content.
So if there is not enough replenishing of CO2 from volcanism (according graphs there currently isn’t), we are all heading to low CO2 low O2 atmosphere.
And now the point. Imagine Earth atmosphere with 170ppm of CO2 and 10% of O2. Atmospheric pressure is 11% lower. Air is still breathable, land creatures living, but not thriving. Able to live only at near sea level altitude. Plants choking on lack of CO2, grasslands. Ice Age conditions.
From temperature point of view, temperature on Moon without atmosphere ranges on equator from -183C to 106C. This is around -77C average. Average temperature of Earth tropics is around 25C. This is 102C difference from moon. And by lack of 11% atmosphere we are 11% closer to this state. 11% from 102C is 11.2C This is rough guess by it will not be far from reality. And surprisingly 11.2C is very very close to difference between ice age state and interglacial.
Thinner atmosphere will bring another effect. It will increase both hot and cold extremes of temperature. Difference between daily high and low will be much bigger. So much for hottest days together with coldest days…
So it looks like thermostat theory is correct. Only input parameters will change.

Phil.
August 24, 2015 1:04 pm

Peter August 24, 2015 at 11:15 am
It looks like we are loosing Oxygen. In 15 years we gained 30ppm of CO2, but lost almost 300ppm of Oxygen!

You have your units mixed up, the loss of O2 is in per meg, 300 per meg = 300/4.8 = 62.5 ppm
The CO2 gain in the atmosphere represents about half of the CO2 created by combustion so the loss of O2 is about what is expected.

Peter
August 24, 2015 2:34 pm

Yes Phil. you are right. there is 21% of O2 in atmosphere so 0.21 x 0.0003 is 0,000063 this roughly represents ppm change in O2 content.
But this is not changing principle I wrote about. That we are (almost) irreversibly losing Carbon and Oxygen sequestrated as CaCO3 in oceans.
By this rate we will lose half of oxygen content in 7900 years..
We are dependent on replenishing of CO2 by volcanism and subsequent transformation of part CO2 to O2 by plants.
Frankly we are dependent of how much Limestone is currently cooking in magma chambers of volcanoes. And we can be sure there are cycles in amount of stored Carbonates in sea floor entering continental shelf subduction zone and coming to contact with Earth Mantle

Phil.
August 25, 2015 7:36 am

Peter August 24, 2015 at 2:34 pm
Yes Phil. you are right. there is 21% of O2 in atmosphere so 0.21 x 0.0003 is 0,000063 this roughly represents ppm change in O2 content.
But this is not changing principle I wrote about. That we are (almost) irreversibly losing Carbon and Oxygen sequestrated as CaCO3 in oceans.

It is reversible, CO2 in the ocean is reversibly linked to pCO2 in the atmosphere.

Steve P
August 24, 2015 1:20 pm

Interesting ideas, but I thought oceans release CO₂ as they warm.

Steve P
August 24, 2015 1:24 pm

Peter wrote: “We are solely dependent on volcanism returning CO2 back to atmosphere.”

August 24, 2015 11:50 am

Mike
August 24, 2015 12:04 pm
August 24, 2015 12:06 pm

August 24, 2015 12:13 pm

Willis says: “I’ll start with the question of the two lags, and I’ll use his terminology from his math. The first lag is the 90° lag in a sinusoidal system between the change in forcing Delta F (∆F) and the thermal response Q. ”
I think you are still misusing the terminology here, Willis. A ‘forcing’ is itself any change from the equilibrium conditions.
Let’s call the long-term average intensity of the incoming sunlight I_avg (@ 1 AU, measured in W/m^2). This has a variation with a period of ~ 11 years which is (approximately) sinusoidal. If that variation has an amplitude of I_o, then we can write the overall intensity as
I = I_avg + ∆I = I_avg + I_o sin( 2 pi t/11 year)
In this equation, ∆I is the “forcing” = the change from equilibrium. We can trivially change the letters from I_o & ∆I to F_o and ∆F. So ∆F *IS* the forcing, not the “change in forcing”.

Mike
August 24, 2015 12:40 pm

I think you’re making the same point I made above. The delta F is the “anomaly” not the rate of change.

August 24, 2015 12:48 pm

Willis, there is one major difference between your analysis and the systems you are trying to model. In your very first example, you have a heater with constant *temperature*, not constant *heat*. In other words, your stove itself can supply large heat inputs when the pan is cold, and that heat input drops to zero as the pan approaches the temperature of the stove.
But the sun does not act this way. The oceans do not approach the temperature of the sun and therefore stop warming any further. The sun does not “supply extra forcing” when the oceans are cool nor drop its heat input simply because the oceans have warmed a bit.
Or put another way, you are modelling a pan moved from a cool room to a hot room, but the actual system is more like a pan in a cool room with a heater than is turned on with constant power. (it gets more involved with sinusoidal inputs, but if you start with a bad model for a step-function input, you will most likely end with a bad model for a sinusoidal input.)

gbaikie
August 24, 2015 2:35 pm

–Willis, there is one major difference between your analysis and the systems you are trying to model. In your very first example, you have a heater with constant *temperature*, not constant *heat*. In other words, your stove itself can supply large heat inputs when the pan is cold, and that heat input drops to zero as the pan approaches the temperature of the stove.
But the sun does not act this way. The oceans do not approach the temperature of the sun and therefore stop warming any further. The sun does not “supply extra forcing” when the oceans are cool nor drop its heat input simply because the oceans have warmed a bit. —
I would say sun does act this way particularly with ocean.
Low flame for a pot of water that doesn’t get warm enough to simmer with the lid on, but don’t
put the lid on. So that is related to the ocean not reaching sun temperature [which below the atmosphere is about 80 C].
another aspect is the sunlight is “diffusing” into the ocean depths so there is more of disconnect
between surface type heating of pan or ground surface warming. The ocean is more similar to heating something with microwave oven [gallons of water in microwave oven].

Ian H
August 24, 2015 3:59 pm

Willis’ analysis is fine as long as you understand that what he calls “input” is really a graph of the equilibrium response if you were to stop the variation responsible for the change you are measuring. His fire and pan analogy is just a simplification to make it easier to understand.
I prefer to think in terms of an electric element rather than a fire so we can imagine turning the knob up and down. The “input” is not the temperature of the element. It is the temperature that the pan would eventually attain if we stopped fiddling with the settings and left it the same for a long period. That is going to be pretty close to the temperature of the element but it won’t be exactly the same due to heat radiated from the pan to the room.
In the case of a graph of temperature during the year, the “input” is the temperature we would see if we could magically stop the march of the planet in its orbit and have it be perpetually August the 25th for long enough for the earth to fully adjust.
In the case of a graph of daily temperature the “input” would be the temperature we would see if we could stop the spinning of the planet. Note that in most cases this will be a much more drastic “input” curve than the yearly case. Staying 2pm forever would have a much greater impact on temperature than staying August forever would.

Curt
August 24, 2015 7:33 pm

Tim: You say to Willis, “In your very first example, you have a heater with constant *temperature*, not constant *heat*. In other words, your stove itself can supply large heat inputs when the pan is cold, and that heat input drops to zero as the pan approaches the temperature of the stove.”
I don’t see that at all. I see an example that has a constant heat input (for a time interval) to a system that can output heat to a colder ambient. The higher the temperature of the system, the greater the losses to ambient. In this simple first-order system, if the constant heat input were applied indefinitely, the temperature of the system would stabilize at a level where its losses to ambient matched the heat input.
So in this basic respect, it is indeed like the sun’s (roughly) constant power input to the earth.

Frank
August 24, 2015 12:54 pm

Willis wrote: “Now, what are we measuring with the thermal transfer coefficient lambda? We are measuring what the thermal response of the system would be if there were no thermal lag. But if there were no thermal lag … then it must be an equilibrium thermal transfer coefficient, or quite close to one. This is interesting because this implies that the lambda values for the no-thermal-lag condition must be close to what is called the “equilibrium climate sensitivity”. ”
When you use dT = lambda*dF in equation 2, you are using an equation that is valid only when the temperature has reached a new EQUILIBRIUM in response to a FORCING. There is no equilibrium during seasonal warming and cooling. You can’t apply formulae appropriate for equilibrium to data from non-equilibrium situations! Not every calculation that yields an answer with units of K/(W/m2) gives ECS.
Without the correct physics, you’ll never get the right relationships. The RATE of warming or cooling of the planet depends on the size of the radiative IMBALANCE and its heat capacity. A radiative forcing is not the same thing as a radiative imbalance. When the surface is warming in response to a fixed forcing, the imbalance will shrink exponentially.
The heat capacity of the air is 10,500 kJ/m2/K. The heat capacity of the ocean is 4185 kJ/m3/K times the depth of the mixed layer in meters (126,000-210,000 kJ/m2/K for 30-50 m). The heat capacity of land may be similar to that of air.
Radiative imbalance is post-albedo incoming SWR minus OLR, which is available from CERES. Otherwise, outgoing OLR depends on surface temperature and fast feedbacks: Planck, WV, LR, cloud, and seasonal snow cover. (Slow feedbacks – ice caps become important over decades and longer.) The sum of these feedbacks is called the climate feedback parameter, which is the reciprocal of ECS.

Ian H
August 24, 2015 1:17 pm

The fact that the maximum for the lagged response lies exactly on the curve for the “fire under the pan” is easy to see.
When the fire is hotter than the pan, the pan will be warming up, so the slope of its temperature graph will be positive. When the pan is cooler than the fire the pan will be cooling down and the slope if its graph will be negative. Any place where the slope switches from positive to negative is by definition a maximum (in calculus this is sometimes called the “first derivative test”), and this can only happen where the temperature of the fire changes from being greater than to being less than the temperature of the pan. So the maximum temperature of the pan will always occurs at a place on the graph where the temperature of the pan and fire are the same.
This is true regardless of the shape of the shape of the “input” – it doesn’t just hold for sinusoidal curves.

August 24, 2015 1:24 pm

Whatever is a cause of the global surface temperature change, it must happen through the temperature change of the ocean and the land. Because the oceans cover 70 % of the Earth’s area, the temperature change depends mainly (= 70 %) on the temperature of the mixing layer (about 75 m deep). The interesting points in the analysis of Willis are the time constant 3 months for the ocean and 1 month for the land. They are very close to the results analysed by Kauppinen et al.: 82 days and 31 days and they are the same as introduced in the comments of Bill Illis.
Here is a connection to the transient climate sensitivity (CS), because IPCC says that the surface temperature reacts less than in a year for the climate forcing at the top of the atmosphere. According to these time lags, this conclusion is correct. IPCC runs into troubles in the temperature pause case. One explanation has been that “the heat has gone into the deep ocean”. This is a pathetic explanation, because it is not possible without the warming of the mixing layer. If it would be possible, the engineers would like to know, how to do it.
I have analysed the time lags between the different sky conditions. For me the most interesting finding is, why the surface temperature of the cloudy sky is 0.4 degrees higher than that of the clear sky. It is a question about the time lags. If you are interested in seeing the explanation, you find the link on this web page:

August 24, 2015 1:28 pm

Sorry that I missed the address. Another try:

Paul Tatman
August 24, 2015 2:51 pm

Willis, I noticed you had a math teacher named Hegji ( Was his first name Ronald?) so did I in the middle 60’s. He also worked for Clair Hill which eventually became CH2M Hill. The more interesting thing about him was the ability to write a sentence with both hands on the chalkboard, left hand doing the first part and his right doing the latter part, in multiple rows continuously. Only person I have ever seen do so.
I really enjoy your work, as a degreed Oceanographer I have been amused by the lunacy of the GW crackpots. Thanks for a little reality.
And thank you Anthony for your weather reports that got it right, I watched and listened for years, then moved away in 2008, followed the blog since near the beginning, there is always a lag in every system.
Paul

John M. Ware
August 25, 2015 1:50 am

I was going to be the first to bring up a certain point, but Ian H has anticipated part of it. I was going to say that I assumed Willis was measuring change in temperature of a pan over a gas stove. Since I have an electric stove, my observations would have some differences. I would never let an electric element achieve its highest temperature uncovered (waste of energy, danger from invisible heat source since, unless temp gets to red-hot, the hot element is the same color as the others). Therefore, I place my pan, with water in it, on the element, then turn it on. For some foods, I wait until the water boils to add them to the water (poaching an egg, for instance); for others, such as beets, I place them in the water and let them heat with the water. But the lag in this case is two-fold: the lag between turning on the heat and actually having it on full force (or to whatever level the knob is set at); and the secondary lag between cold water and water warmed by the now-warm pan over the now-warm stove element. I know, it’s an added element of complexity with which Willis’s excellent but abstract article does not wish to deal; but the concept is interesting to me.

TonyN
August 25, 2015 6:14 am

Don’t know if this has been covered, but might Willis be interested in looking at the Mauna Loa CO2 graph annual oscillation?