Learning by Seeing

Everyday you may learn something new. I just discovered that I made a mistake in the recent post on IR-detectors believing that an optical lens may increase the radiance from a target onto a detector, because a (positive) lens increases the amplitude of the incoming light waves by making the rays converge.

But then I missed that the viewing angle also increases which makes the radiance the same after the lens as before. This means that the detector even with an optical lens cannot reach a higher temperature than the target, in full accordance with the 2nd law.

Climate Crisis vs Financial Crisis?

CO2 alarmism is based on the Kiel-Trenberth global energy budget with massive two-way transfer of heat energy between the atmosphere and the Earth of the order of 350 W/m2 as Downwelling/Upwelling Longwave Radiation (DLR/ULR), about the same as the total input from the Sun.

There is an analogy in global economy as a massive two-way transfer of money between state and people, or between central banks and commercial banks. The financial crisis can be seen as an expression of the instability arising from such a massive two-way transfer, which appears to create heaps of money out of nothing, like writing 0 = 100 - 100 and seeing 100s appearing from 0.

DLR/ULR similarly represents an instability of global climate, which by CO2 alarmists is being used to sell a climate crisis.

The financial crisis is real, because there is an unstable two-way flow of money between state and people. The question is now if the climate crisis is real, because there is an unstable two-way flow of heat energy between the atmosphere and the Earth as DLR/ULR?

In short: Is DLR/ULR reality or fiction? Does Nature have a central bank which can issue energy bonds and supply a global climate market with energy capital, out of nothing?

No, there is no direct evidence that Nature has such a central bank. There is no direct evidence that DLR/ULR is real, and thus there is no direct evidence of a climate crisis.

GWPF reports Europe Reconsidering Its Unilateral Climate Policy. It is clear that Europe cannot handle both a financial crisis and a climate crisis. If there is no climate crisis after all, then the financial crisis may be possible to resolve by limiting the unstable two-way transfer of money, following what Mother Earth is doing with the flow of energy to keep a stable climate.

Evidence that DLR is Fiction?

The evidence of Downwelling Longwave Radiation put forward by CO2 alarmists is that an IR-detector of some sort when pointed to the sky gives a reading which is not zero, and the non-zero reading is used as evidence that something has been recorded, which is then interpreted as DLR or backradiation from the cold sky to the warmer surface of the Earth.

The evidence of DLR is thus a non-zero reading of an IR-detector.

Suppose now, to turn the argument around, that the reading of the IR-detector is zero. That would be the case if the IR-detector works by detecting a temperature difference and if the temperature difference happens to be zero. This would be the case if the temperature of the Earth is equal to the temperature of the sky. The IR-detector working on a principle of

temperature difference, like a thermocouple, would then record zero, as if nothing was recorded.

The following questions present themselves:

• What could be concluded from a zero reading of an IR-detector?
• That there is positive DLR?
• That there is positive DLR precisely balanced by Upwelling Longwave Radiation?
• Or can nothing positive be concluded from a zero reading?

IR Detectors and True-SB

In the previous post From Correct Planck Law to True-SB and False-SB I derived the following Stefan-Boltzmann Law based on the new proof of Planck's radiation law presented in Computational Blackbody Radiation in Slaying the Skydragon:

(True-SB) $R(T,T_b) =\int_{T_b}^{T}\gamma T f^2 df + \int_0^{T_b}\gamma (T - T_b) f^2df \equiv I_1 + I_2$,

where $R(T,T_b)$ is the radiance from a blackbody of temperature $T$ into a blackbody background of temperature $T_b < T$.

Here the first integral $I_1$ is the radiance from the blackbody absorbed as heat by the background above the cut-off of the background and $I_2$ the net heat absorbed below cut-off as the difference between absorbed and emitted radiation.

Let us see what (True-SB) can tell us about the possible design of IR detectors, where $T$ is the temperature of the object to be detected and $T_b$ that of the detector. There are two types of detectors to consider (see here):

1. cooled with $T_b < T$
2. un-cooled with possibly $T_b > T$.

In a cooled detector with $T_b < T$ both $I_1$ and $I_2$ are positive and $R(T,T_b)$ can

be directly recorded as heating transferred into an electric signal by a thermocouple. (For a star a large distance only $I_1$ is recorded, with the help of optical magnification of the input,

but this is not IR.)

In an un-cooled detector the heat transfer would be away from the detector if $T_b>T$, and

in principle the object could be detected from recording the cooling of the detector. But this is

a form of negative detection which may be difficult to get to work in practice.

More interesting is to explore what (True-SB) suggests for the design by heating in the case the object has lower temperature than the detector:

In this case the signal from the object does not contain frequencies above the cut-off of the detector and so $I_1=0$, but by magnifying the signal optically (by a lense) the radiation below cut-off can be changed into

• $I_2(M)= \int_0^{T_b}\gamma (MT - T_b) f^2df$

where $M>1$ is a magnification factor and $I_2(M)$ may thus become positive even if $T_b > T$.

We sum up the experience from (True-SB):

1. If the temperature of the detector is lower than the object, then direct recording of heating is possible (with $I_1>0$ and possibly also $I_2>0$ ).
2. If the temperature of the detector is higher than that of the object, then heating may be recorded after optical input magnification (with $I_2>0$).

Note added: I made a mistake believing that a lens can magnify radiance, which I correct in Learning by Seeing.

S Vill Satsa på IT i Skolan

Jag har nu fått följande respons från S beträffande matematik och IT i morgondagens skola:

• Hej Claes!
• Socialdemokraterna stödjer en investering i matematik men vi vet ännu inte hur regeringen vill lägga upp programmet och kommer självklart ha synpunkter på hur så bör ske. Vad gäller en modernisering av skolan för att passa it-samhället så har vi har under flera års tid kritiserat regeringen för att inte ta informationsteknologin på allvar, vi vet att detta skulle kunna ha stor betydelse för t ex undervisningen i matematik.
• Vi skulle gärna ta del mer av hur du tycker att matematikundervisningen och it iskolan bör utvecklas och undrar om du har möjlighet att träffa Socialdemokraterna i riksdagens utbildningsutskott onsdagen den 30 november kl 14.00-15.30. Inbjuden till detta tillfälle är också Trevor Dolan för att ge sin syn på matematik och kreativitet.
• Med vänlig hälsning, Kristina Persdotter.

Äntligen, kan man väl säga. S förstår någonting mycket viktigt som Björklund inte alls förstår. IT kommer att revolutionera lärande på alla nivåer. Inom ett par år, då alla elever har en iPad.

Denna fråga kan lyfta S ur sitt svarta hål. Det skall bli mycket intressant att se om S är förmöget att utnyttja den möjlighet att ta initiativet från oppositionen som Björklund nu serverar med sin "katederundervisning" utan IT och framtid.

Rapport inför och efter mötet den 30 november kommer så småningom.

From Correct Planck Law to True-SB and False-SB

In recent posts I have exhibited a version of Stefan-Boltzmann's radiation law referred to as False-SB which is used by climate scientists to support CO2 alarm, and I have shown that False-SB is the result of an incorrect application of Planck's radiation law. Let me here again show the incorrect argument leading to False-SB, and the correct argument leading to

True-SB.

Stefan-Boltzmann's radiation law in its original form (SB) expressing the total radiance from a blackbody of temperature T into a background at 0 K, is obtained by integration/summation over frequencies of Planck's radiation law expressing the radiation intensity $I(f,T)$ emitted by a blackbody as a function of frequency $f$ and temperature $T$, per unit frequency, surface area, viewing solid angle and time:

• $I(f,T)=\gamma Tf^2\theta (\nu, T),\quad \gamma =\frac{2k}{c^2}$

with the high-frequency cut-off factor

• $\theta (f, T)=\frac{\frac{hf}{kT}}{e^\frac{hf}{kT}-1}$,

where $c$ is the speed of light in vacuum, $k$ is Boltzmann's constant, with $\theta (\nu ,T)\approx 0$ for $\frac{hf}{kT}>10$ say and $\theta (f ,T)\approx 1$ for $\frac{hf}{kT}<1$. Since $h/k\approx 10^{-10}$, this effectively means that only frequencies $\nu\le T 10^{11} $ will be emitted, which fits with the common experience that a black surface heated by the high-frequency light from the Sun, will not itself shine like the Sun, but radiate only lower frequencies.

We refer to $\frac{kT}{h}$ as the cut-off frequency because frequencies $f>\frac{kT}{h}$ will be radiated subject to strong damping. We see that the cut-off frequency scales with $T$, which is Wien's Displacement Law.

Normalizing and simplifying the exponential cut-off, Planck's law can be written in the form

• $I(f ,T) =\gamma Tf^2$ for $f\le T$
• $I(f,T) = 0$ for $f > T$.

The idea is now that SB is obtained from Planck's law by summation/integration over frequencies. The basic form of SB expresses the total radiance $R(T,0)$ of a blackbody of temperature $T$ radiating into a background at 0 K as

• $R(T,0)=\gamma T\int_0^{T}f^2df = \sigma T^4$

where $\sigma =\frac{\gamma}{3}$.

We next seek the radiance $R(T,T_b)$ when the background is a blackbody of temperature $T_b > 0$. Planck proved his law in the case $T_b=0$ by using an argument based on statistics of quanta and it is not clear how to extend this argument to $T > T_b > 0$. Accordingly, a derivation of $R(T,T_b)$ from Planck law in the case $T_b > 0$, appears to be missing in physics literature.

To be able to compute $R(T,T_b)$ a new proof of Planck's law was given in Computational Blackbody Radiation in Slaying the Skydragon, a proof which allows direct generalization to $T_b>0$ with (compare with previous post):

• $R_{True}\equiv R(T,T_b) =\int_{T_b}^{T}\gamma T f^2 df + \int_0^{T_b}\gamma (T - T_b) f^2df \equiv I_1 + I_2$,

where the first integral $I_1$ is the radiance from the body into the background above the cut-off of the background and the second integral the net radiance below cut-off. Notice that if $T\approx T_b$, then $I_1\approx 3 I_2$ and so the above cut-off contribution $I_1$ dominates the radiance.

We see that $R(T,T_b}$ is the sum of two integrals with positive integrands both expressing radiance from the warm body into the colder background. We should now stop here having reached an expression for $R_{True}=R(T,T_b)$ correctly derived from a correct Planck law.

However, climate scientists have introduced an incorrect version of SB named False-SB, obtained by rewriting $I_1$ as follows:

• $I_1 = \int_0^{T}\gamma Tf^2df - \int_0^{T_b}\gamma Tf^2df =\sigma T^4 - \sigma TT_b^3$

which since $I_2 = \sigma (T-T_b)T_b^3$ gives False-SB on the (seductively simple) form

• $R_{False} = \sigma T^4 - \sigma T_b^4$

expressing the transfer of energy from the body to the background as the difference of two gross flows in opposite directions. We see that False-SB arises by rewriting an integral with positive integrand as the difference of two integrals with positive integrands as follows:

• $\int_{T_b}^{T} f^2 df = \int_0^{T}f^2df - \int_0^{T_b}f^2df$,

where the lower integration limit $0$ could be replaced by any positive number smaller than $T_b$. False-SB arises when giving this formal mathematical manipulation a physical meaning stating that one-way net flow is the difference of two-way gross flows, with the flow from the background in violating with the 2nd law of thermodynamics.

We see that False-SB arises by writing the correct $R(T,T_b)$ as

• $R_{False} = (R(T,T_b) + C) - C$,

where $C$ is an arbitrary positive constant, and then assigning the arbitrary constant $C$ a definite physical meaning as the transfer of heat from the colder background to the warmer body, in violation of the 2nd law of thermodynamics. The only correct choice is $C=0$ which gives the correct net flow $R(T,T_b)$ as stated above.

False-SB is thus obtained by an ad hoc generalization of SB for $T_b=0$ in the form $R(T,0)=\sigma T^4$, to the incorrect form $R_{False} = \sigma T^4 - \sigma T_b^4$ in striking violation of the 2nd law.

Note that the seduction of $R_{False}$ is enhanced by the fact that $R_{False}$ gives the

correct net flow $R(T,T_b)$, which by proponents of the correctness of $R_{False}$ is used as evidence that $R_{False}$ is correct. But this only shows that there is an aspect (one-way net flow) of $R_{False}$ which is correct, while the two-way gross flow suggested by $R_{False}$ is grossly incorrect.

If the only meaning of $R_{False}$ is net flow equal to $R(T,T_b)$, then $R_{False}$ should better be eliminated from the discussion altogether by replacing it with the correct $R(T,T_b)$. DLR should thus be eliminated as fictitious pseudo-physics.

Finally, the difference between $R_{True}$ and $R_{False}$ comes out as different stability

properties of one-way net flow and two-way gross flow, with $R_{False}$ supporting the idea of high climate sensitivity behind CO2 alarmism. If $R_{False}$ and DLR is eliminated from the discussion, then CO2 alarmism crumbles.

Curry on Straw Dog of False-SB

Judy Curry answers on her blog my question about Stefan-Boltzmann's radiation law and DLR:

• Claes, true to form, you set up a straw dog to knock down. The Stefan Boltzmann equation is the integral of the Planck function.
• No one calculates atmospheric radiative heat transfer using the Stefan Boltzmann equation (not since the 1960′s, anyways).

I follow up with the following questions related to Judy's statement that DLR/False-SB is a straw dog:

1. Do you agree with me that DLR should once and for all be removed from climate science?
2. Do you agree with me that the Kiehl-Trenberth energy budget with DLR is grossly incorrect?
3. Do you agree with me that climate scientists have invented a form of SB which does not have support in physics literature, and which in fact is false because it involves a violation of the 2nd law?
4. Have you read an digested my derivations of Planck/SB without statistics of quanta?I appreciate prompt answers, which should be possible since you are now on line.

I will report on Judy's answers...It appears that Judy is thinking deeply about what to say and

this may take time, but the questions remain and will not disappear just because Judy closes her eyes...

False-SB Violates the 2nd Law

To understand the difference between the two versions of Stefan-Boltzmann's radiation law (SB) under discussion, True-SB and False-SB, let us inspect the proof of SB from Planck's radiation law in its normalized form presented in Computational Blackbody Radiation:

• $R(f ,T) =\gamma Tf^2$ for $f\le T$
• $R(f,T) = 0$ for $f > T$

where $R(f ,T)$ is the radiance of frequency $f$ from a blackbody of temperature $T$, $\sigma$ is a constant and a simplified high-frequency cut-off is used (as compared to Planck's exponential cut-off).

The total radiative transfer $R_{True}$ to a blackbody 1 of temperature $T_1$ from a blackbody 2 of temperature $T_2>T_1$ is given by integration over frequencies as follows:

• $R_{True} =\int_{T_1}^{T_2}\gamma T_2 f^2 df + \int_0^{T_1}\gamma (T_2 - T_1) f^2df \equiv I_1 + I_2$,

where the first integral $I_1$ is the heating effect from 2 above the cut-off of 1 and the second integral the net heating from 2 below cut-off.

We see that $R_{True}$ expressing True-SB is the sum of two integrals with positive integrands, that is, $R_{True}$ is the sum of many small positive contributions all with transfer of heat from 2 to 1.

We shall now see that False-SB arises by rewriting $I_1$ as follows:

• $I_1 = \int_0^{T_2}\gamma T_2f^2df - \int_0^{T_1}\gamma T_2f^2df =\sigma T_2^4 - \sigma T_2T_1^3$

where $\sigma =\frac{\gamma}{3}$, which since $I_2 = \sigma (T_2-T_1)T_1^3$ gives False-SB on the form

• $R_{False} = \sigma T_2^4 - \sigma T_1^4$

expressing the transfer of energy from 2 to 1 as the difference of two gross flows with different signs. We see that False-SB arises by rewriting an integral with positive integrand as the difference of two integrals of different signs as follows:

• $\int_{T_1}^{T_2} f^2 df = \int_0^{T_2}f^2df - \int_0^{T_1}f^2df$,

where the lower integration limit $0$ could be replaced by any positive number smaller than $T_1$. False-SB arises when giving this formal mathematical manipulation a physical meaning stating that one-way net flow is the difference of two-way gross flows, with the flow from 1 to 2 violating the 2nd law of thermodynamics and involving an arbitrary constant.

False-SB thus arises by an ad hoc translation to physics of a mathematical operation which results in a violation of the 2nd law of thermodynamics.

Accordingly False-SB is not found in physics literature, but has appeared outside physics as an ad hoc free invention by climate scientists for the purpose of selling CO2 alarm.

Stability of True-SB vs False-SB

The different stability properties of the two versions discussed in recent posts of Stefan-Boltzmann's radiation law, True-SB and False-SB, can be exhibited by comparing the stability of the following simplified SBs:

1. $R = T_1 - T_2$
2. $R = T_3$ where $T_3 = T_1 - T_2$,

where $R$ is the heat flow between two bodies of temperature $T_1$ and $T_2$, expressed in two different forms: 1. as the difference of two-way gross flows and 2. as one-way net flow.

Stability concerns the effect on the output $R$ from perturbations of input ($T_1$, $T_2$ and $T_3$). Denoting the perturbation of $T_i$ by $dT_i$ for $i=1,2,3$, it is natural to assume a bound $E$ on relative perturbation of the form

• $\frac{\vert dT_i\vert}{T_i}\le E$ for $i=1,2,3$.

Estimating the effect $dR$ on the output $R$ from the $dT_i$, we have

$\vert dR\vert \le \vert dT_1\vert +\vert dT_2\vert = \frac{\vert dT_1\vert}{T_1} T_1 + \frac{\vert dT_2\vert}{T_2} T_2 \le ET_1 + ET_2=E(T_1+T_2)$,

which is to be compared with

$\vert dR\vert \le \frac{\vert dT_3\vert}{T_3}T_3 \le E T_3$.

If $T_3$ is much smaller than $T_1+ T_2$, then the second bound is much smaller which expresses that 2. is more stable than 1. or that 1. is more unstable than 2.

A particular case is given by $T_1=T_2$ with $T_3 =0$. This is the case of two blackbodies of equal temperature which according to 2. is very stable, but according to 1. much less stable.

The different stability gets expressed in different assessments of climate sensitivity, from 1. unstable with alarm to 2. stable without alarm.

Question to Curry about Stefan-Boltzmann and DLR

In a recent post Judy Curry lets Grant W Petty speak out with reference the ongoing discussion on my blog with Petty. Curry assists Petty with:

• I will continue to (barely) follow Claes Johnson’s work to see if he is able to come with anything interesting or publishable.

Here is something interesting Judy of main importance to CO2 alarmism: In recent posts on Downwelling Longwave Radiation DLR and Stefan-Boltzmann's radiation law (SB) I have shown that CO2 alarmism is based on a version of SB without support in physics literature, a version I have referred to as False-SB, and I have compared False-SB with a correct version named True-SB.

Since you take up my work on your blog in critical terms, I think I have reason to ask you to articulate your critique of my work and in particular state if you consider False-SB to be true with a solid support in theoretical physics and experiment. Is it so? Or do consider my critique to be well founded?