Photon Foolishness and CO2 Alarmism

Einstein received the Nobel Prize in Physics in 1921 for his 1905 discovery of the Law of Photoelectricity (discovered by Hertz already in 1887) based on an idea of light as a stream of light particles or light quanta later named photons, in a return to an idea of Newton abandoned since the discovery of light as an electromagnetic wave phenomenon captured by Maxwell's equations published in 1873.  

Einstein was not happy with the Prize motivation, since it explicitly stated that he was not awarded because of his theory of relativity, which he considered to be his main work, while he viewed his early work on photoelectricity rather as a misconception, since concerning photons/light quanta he confessed in 1951:

• All these 50 years of conscious brooding have brought me no nearer to the answer to the question, "What are light quanta"? Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.

Unfortunately, the Tom, Dick and Harry misconceived idea of light as a stream of photon particles has survived into our days, in parallel with the wave picture, and has come to serve as the basis of CO2 alarmism in the form of Downwelling Long Wave Radiation DLWR of Back Radiation as a stream of photons from the atmosphere to a warmer Earth surface with a massive global warming effect. 

In the spirit of Bohr the particle and wave nature of light are not considered contradictory but simply complementary although behaving differently:  

The Tom, Dick and Harry particle misconception is captured in an incorrect Planck-Stefan-Boltzmann Law PSBL stating that a black body at temperature $T$ Kelvin emits/radiates heat energy in the form of light quanta/photons scaling with $T^4$ (per unit area and time), independent of the surrounding temperature. The radiation has a Planck spectrum scaling with $T\nu^2$ with $\nu$ frequency (modulo high-frequency cut-off scaling with $T$). The misconception is that the radiation is independent of the surrounding temperature based on a primitive idea of radiation as a stream of photon particles being ejected independent of surrounding. This misconception is widely spread and embraced by otherwise very knowledgable physicists and laymen. 

A correct PSBL states black body radiation scaling with $(T^4 - T_s^4)$, where $T_s$ is the surrounding temperature. In this form the radiation can be seen as a wave resonance phenomenon between black body and surrounding, see Computational Blackbody Radiation. 

The Planck spectrum scaling with $T\nu^2$ directly connects with the wave nature of light with the energy of a harmonic oscillator of frequency $\nu$ scaling with $\nu^2$. 

To fit this into a particle idea Einstein suggested to view a photon as a localised wave packet of length scaling with $\frac{1}{\nu}$ and energy scaling with $\nu$ (captured in Planck's formula $E=h\nu$ with $h$ Planck's constant). The total radiation from a a stream of photons would then scale with $\nu^2$ since $\nu$ photons of length $\frac{1}{\nu}$ (traveling with the speed light) would pass in unit time. 

Einstein thus in 1905 associated the energy $E=h\nu$ to a concept of light quanta, which gave him the Nobel Prize in 1921 with the Law of Photoelectricity taking the form $E+P=h\nu$ with $P$ electron release energy and $E$ kinetic energy of an emitted electron upon impact by one photon with energy $h\nu$, but then misled generations of physicists into a misconception of PSBL misused by CO2 alarmism, while his insight in 1951 that light quanta has no physical meaning passed by without notice.  

This post directly connects to the following recent posts:

• Discussion with Will Happer on Back Radiation 
• Radiation as Resonance     

and to a wave analysis of the photoelectric effect (p 97). The idea of light as a stream of photon particles is as misconceived as an idea of sound as a stream of phonon particles which you spit out when you speak, while we all know that sound is transmitted by sound waves as a resonance phenomenon from loud speaker to your eardrums carried by air. 

The idea of light from Proxima Centauri as the closest star to our own as a stream of photon particles traveling at the speed of light one by one all alone 40,208,000,000,000 km on a journey taking 4.37 years without ever getting lost in cosmic dust or atmosphere until finally being captured by a human eye, is to fantastic to be credible. Light as particles is not physics, as Einstein said.

PS Typical misconception of photon particles each one ejecting an electron thus creating photoelectricity:

Compare with Mathematical Physics of Blackbody Radiation describing instead photoelectricity as a wave threshold phenomenon asking for a high enough frequency for electron ejection. See also this post.

Skall Sverige Rädda Tyskland från Kall Dusch?

Svenska Dagbladet meddelar idag svenska folket:

• Medan politiker i Sverige tävlar om att sänka hushållens elräkningar, tävlar politikerna i Tyskland om att duscha kallt och fort för att dra ner på elkonsumtionen så att naturgasen ska räcka över vintern.

SvD har intervjuat Erik Ek, strategisk driftschef på statliga Svenska Kraftnät, som säger:

• Krismedvetenheten är låg i Sverige jämfört med hur det ser ut i Europa, fast vi pratar så mycket om elpriserna.
• Tyskar, spanjorer och nederländare vet exakt, rent praktiskt, vad de ska göra för att vintern ska klaras bättre.
• Där har man ministrar som säger hur länge de duschar för inte det inte ska gå åt så mycket energi för att värma duschvattnet. Kommunala bad stänger av varmvattnet i duscharna och fontäner pumpas inte runt i vissa tyska städer. 

Erik Ek utvecklar vidare:

• Om vi skulle börja spara el nu och i stället exportera ännu mer el till Tyskland, så kan Tyskland spara naturgas i bergrummen.

Sverige skulle alltså med hjälp av Svenska Kraftnät, om nu alla vi svenskar sparar duktigt under återstoden av den svenska sommaren och nöjer oss med friska havsbad (gärna långt in på hösten) istället för varma duschar, kunna säkerställa att tyska ministrar under kommande vargavinter kan duscha om inte varmt så i alla fall inte iskallt. Vill vi ställa upp? Vi är väl fortfarande en humanitär stormakt?

PS Man kan väl säga att vad gäller dusch så passar kallt och fort väl ihop, men å andra sidan skulle man ju kunna njuta av långa kalla duschar eftersom ingen uppvärmning fordras, så varför inte passa på så länge det finns vatten?

 

Logarithmic Effect of More CO2 without Theoretical Basis

To serve as a basis for CO2 alarmism (together with back radiation), IPCC put together the following formula with inspiration from the Beer-Lambert law (exponential transport decrease in an absorbing medium):

• $\Delta RF = 5.35\ln (C/C_0)$ 

where $RF$ is Radiative Forcing from CO2 as a greenhouse gas, $\Delta RF$ is additional forcing caused by a change of concentration of atmospheric CO2 from $C_0$ (preindustrial) to a present $C$. Doubled concentration from preindustrial level would then cause an additional (warming) forcing of about $4$ Watts/m2, which is translated to 1C warming by Stefan-Boltzmann, which by feed back can become anything you like 1-5C with 5C utterly alarming! Note that without the 1C from the formula, feedback has nothing start from and so it is an absolutely crucial element of CO2 alarmism!

In CALCULATION OF ATMOSPHERIC RADIATIVE FORCING (WARMING EFFECT) OF CARBON DIOXIDE AT ANY CONCENTRATION by H. Douglas Lightfoot and Orval A. Mamer an effort is made to improve the formula from the starting point:

• Even though there is no theoretical basis for the Beer-Lambert formula, ∆RF = αln(C/Co), it has been accepted by the scientific community as a reasonable approximation.

but somewhat disappointingly ending up with:

• In this paper we propose an improved mathematical approximation that, like the Beer-Lambert law, has no theoretical basis.

Does a formula with no theoretical basis have a serious defect? Not always, since a formula may capture observations as some form of condensed experience. The validity of the formula can then be checked against observation/experience, and so be falsified. 

However, if there is no way to check the validity of the formula by observation, and this is the case with the above formula, then lack of theoretical basis is a serious defect because theory is what remains if you take away observation. 

So IPCC relies in its prediction of the warming effect of doubled CO2 on a formula which has neither theoretical basis nor observational support. You can say that this is shaky. How can you proclaim Net Zero with all its devastating consequences from nothing? It seems you must have some hidden agenda. You can learn about the agenda in new best seller The Truth about Energy, Global Warming, and Climate Change: Exposing Climate Lies in an Age of Disinformation by Jerome R. Corsi.

Buying a Tesla vs Pyrgeometer Realities

If you feel depressed, here are two nice ways to get happy again: Buy a Tesla or a Pyrgeometer. You know what a Tesla is and what a pyrgeometer is can be read by downloading the manual for the Kipp and Zonen CGR3 Pyrgeometer. It shows that a pyrgeometer, when directed to the sky, on its display shows Downwelling Long Wave Radiation DLWR (also named back radiation) from a colder atmosphere to a warmer Earth surface of typically size 340 W/m2 to be compared with the 170 W/m2 absorbed by the surface from the Sun, about two extra Suns. 

If you are a CO2 alarmist this makes you really happy because you can now point to these two extra Suns as a supposed massive effect from so called atmospheric greenhouse gasses supposed to radiate massive DWLR. 

Now, the manual shows that the thermopile of the pyrgeometer measures a voltage scaling with the difference of atmospheric and Earth surface temperature of typically 15 C with estimated net radiative flux of 60 W/m2 from the surface to the atmosphere. That is what is measured, which is not so fun to look at for an alarmist, so to make such people happy Kipp and Zonen instead displays 340 W/m2 from the atmosphere to the surface computed from the following equation

• 400 = 340 + 60, 

where 400 is by Planck postulated Outgoing Long Wave Radiation OLWR from the pyrgeometer at 15 C, as if the pyrgeometer as a blackbody is in radiative contact with the cold outer space at 0 Kelvin.  Further, 340 is DWLR and 60 as above radiation from surface to atmosphere. So we get 340 = 400 - 60 as DWLR. But doing so the pyrgeometer acts as a ghost detector by assuming massive OLWR from the pyrgeometer as if it is in radiative contact with cold outer space at 0 C, while in fact it is radiative contact with a 15 C colder  atmosphere. What is measured is 60 up, but what is reported is 340 down. 

This is nothing but scientific fraud created by a misrepresentation of a key physical fact: The pyrgeometer is not in radiative contact with outer space at 0 K, but with a 15 C colder atmosphere (the atmospheric window is small). This is massive fraud serving as the instrumental basis for Net Zero, which if implemented would throw humanity back to the Stone Edge at greatly reduced numbers. Can you think of something bigger?

When you realise this you will get depressed again, but then after a second thought you can restore happiness by recalling that you have revealed/understood the scientific fraud of global warming, and then you can proceed to a happy life without worrying about CO2 emissions and Net Zero. Happy, right?

To compare with the Tesla, suppose your Government offers you a massive repay/refund as Downwelling Government Money DGM of 70.000 dollars if you purchase a new Tesla for 80.000 and thus only have to pay net 10.000 according to the formula:

• 80 = 70 + 10.

You would then get happy, right? But you may quickly get a second thought and ask who will pay the DGM? From where can this money come? Could it be that it will come from taxes you pay, so that in fact you have to pay the full amount 80.000, which is way beyond your budget. Ok, so this will make you depressed. But again, when you realise that there is no need for any Tesla at all if there is no Net Zero, and so you are not pressed to buy a Tesla to save the World. Happy, right?

If you don't think that what I say above is true, take a look at the following Earth's Energy Budget presented by NASA:

Compare now with the Wikipedia energy budget without Back Radiation DWLR:

We thus meet two versions of Earth's energy budget underlying CO2 alarmism, one with and one without Back Radiation. This connects to Bohr's idea of complementarity: Light is both particles (photons) and waves, which are viewed not as contradictory but simply as complementary views of a richer particle-wave phenomenon. In fact, any contradiction in physics can be handled this way, in particular Earth's energy budget, which in a fundamental way is based on Back Radiation (top picture), while at the same time it has nothing to do with any such concept (below picture). Back Radiation is truly fundamental, yet you can do without it completely. It exists and does not exist, and that is no contradiction, only complementary views. This is modern physics at its best. 

If you have in your hands both A and notA as being true, then you can win any discussion. Whatever your opponent say, A or notA, you can say that he/she is wrong and that you are right. Very clever strategy.

Try it to see how smoothly it works!

 

The Distinction between State and Process Variable

The discussion with Will Happer about the pyrgeometer recorded in recent posts illuminates the distinction between state and and process in physics, with process connecting to transition between states. For the pyrgeometer the question is if it primarily/directly measures (a) temperature/state or (b) radiation/process. Happer says (b) and and I say (a), while a pyrgeometer manual states that (b) is computed/derived from (a) and so is not directly measured. 

To get perspective, consider a 3d coordinate system by which we can measure/record the position of an object as a form of state, assuming it does not change position. This seems like a fairly straight forward thing to do. 

Assume now that the body changes position with time, which brings in the notion of time to which we will return, and we then meet the concept of velocity as change of position vs change of time. It then seems natural to view velocity as a process variable since it involves transition between two different positions/states over some (infinitesimally small) change of time. If position is easy to measure, velocity seems to be more difficult since it involves measuring a small change of position over a small change of time. 

So it seems natural to view position as a state variable and velocity as a process variable, with a natural connection of transition to change of time. Translated to the pyrgeometer it suggests that (a) is easy (state) and (b) difficult (process). Accordingly a pyrgeometer can be expected to be designed to measure (a). If simple works, why aim for difficult?

Similarly, conductive heat flux scaling with temperature gradient is a process variable analogous to radiative heat flux/radiation. 

This brings up the question if there is any difference between space variable(s) and time variable? Yes, if it is natural to view position as a state variable, then it may be natural to view time as a process variable as something always in transition form one time instance to the next. 

Thus there seems to be a clear distinction between space (state) and time (process) variable. If so, Einstein's key notion of space-time variable with time acting like a 4th space dimension does not seem to be natural, since it makes time a state variable instead of process variable and if you mess things up then you will have a mess.   

Thermodynamics of Capitalism vs Zero Interest Rate

The 2nd Law of thermodynamics expresses that slightly viscous gas/fluid dynamics includes turbulence as cascade of ordered kinetic motion/energy from bigger to smaller scales which is terminated by dissipation into heat energy as smallest scale unordered kinetic motion/energy. The cascade of turbulence into heat energy puts a limit to the increasing strains building from gas/fluid particles with different velocities brought into contact by slightly viscous convection thus allowing the flow to persist over time without blow up or break down. 

The turbulent dissipation is viewed as a loss of large scale ordered kinetic energy or cost, which is necessary to keep slightly viscous flow alive. It is considered to be a loss since the produced low-quality heat energy as unordered kinetic energy cannot fully be recycled into high-quality large scale ordered kinetic energy, because the precision required is not available. You can burn a book into heat energy and ashes, but you cannot reconstruct the book from the ashes. This is developed in detail in Computational Thermodynamics.

One can make a connection to the Schumpeter's Creative Destruction as the incessant product and process innovation mechanism by which new production units replace outdated ones as an essential fact about capitalism. 

The creative destruction is a loss of value of process and product capital and it seems natural to balance with an interest rate as a loss/cost of financial capital. 

A capitalistic system can then be seen as slightly viscous flow in which large strains can develop because free enterprise as ordered large scale kinetic energy is allowed, and then as a system which needs a certain level of creative destruction balanced by interest rate to keep going without break down. 

In particular, setting the interest rate to zero (ZIRP), would give a system without financial limits prone to break down. This is what we are now witnessing unfold after 20 years of zero interest rate.  

The idea behind ZIRP seems to be to refinance state budgets out of control. Compare with this exposition of the dark side of ZIRP now coming up: It appears that a healthy bond yield/risk premium/interest rate could be 3% + inflation, thus 10% if inflation is 7%. Looks good?  

Note that turbulent dissipation is the way nature puts a limit to the increasing strains/velocity gradients building up in a slightly viscous flow so that the flow can continue/show go on. In a ZIRP capitalistic economy huge wealth gradients develop which put society under increasing strain/inequalities...  

How Much Do You Radiate?

A human body with surface 1 m2 sitting still in a room without air circulation and windows eating a standard ransom of 2.000 kcal/day produces a heat power of 80 Watts (two standard light bulbs), which has to be transferred by radiation to the room walls (no convection/conduction) to keep stationary state and not overheat or freeze. 

Assuming a body surface temperature of 30 C and a wall temperature of 10 C, we can compare with Stefan-Boltzmann's Law stating a radiative flux from body to wall of about 80 Watts/m2 upon a temperature difference of 20 C. This is basic physics. If the wall temperature was 20 C then the heat transfer would be 40 Watts/m2 and so the body would overheat. If the wall temperature is 0 C, then you will freeze to death unless you increase your intake to 3000 kcal/day. This may be the case for the coming Winter.

We are here clearly speaking about one-way heat transport from body to wall. Let us now compare with the center piece of CO2 alarmism, which is two-way heat transfer between a colder atmosphere and a warmer  Earth surface with "back radiation" from the atmosphere to the Earth surface of size 350 Watts/m2 to be compared with the around 200 Watts/m2 absorbed from the Sun. 

This can be compared with the human-room situation in the case of a temperature difference of 10 C corresponding to a net heat transfer of 40 Watts/m2 and would then postulate a radiative heat transfer from wall to body of 350 Watts/m2, to be balanced by outgoing radiation from the body to wall of 390 Watts/m2 with a net of 40 Watts/m2. The two-way heat transfer of CO2 alarmism thus postulates a massive back-and-forth heat energy transfer between body and wall of size 400 Watts/m2 (the power of two Suns) while the net is small 40 Watts/m2. 

Your body is thus supposed to be shining like two Suns which is balanced by the walls also shining like two Suns. Shining like two Suns could boost your ego, but you understand that this can only be fiction, right? If so, then you also understand that the back radiation of CO2 alarmism is also fiction.

Why is massive back-and-forth heat transfer between bodies unphysical? Because it is unstable and as such not physically sustainable over time. Only net heat transfer from warm to cold is stable and thus physically possible and then in fact also real. See Computational Black Body Radiation. 

The unphysical nature of back radiation can be seen in the functioning of a pyrgeometer discussed in previous posts clarifying that a pyrgeometer de facto measures a temperature difference and in no way gross back radiation, which is a derived ghost quantity which cannot be directly measured because it lacks stable physical realisation. Similarly your body can register the temperature of the wall of the room you are sitting in, but has no collector counting photon particles spitted out from the colder wall, nor any sensor counting photons emitted by your body.   

  

Is Radiative Heat Transfer a Resonance Phenomenon Between Bodies?

Computational BlackBody Radiation offers a new proof of the Planck-Stefan-Boltzmann Law PSB based on electromagnetic wave resonance under deterministic finite precision computation, taking the form

• $Q = \sigma (T_A^4 - T_B^4),$       (1)

where $Q$ is (normalised) radiative transfer of heat energy between two blackbodies A and B with temperatures $T_A$ and $T_B$ Kelvin, and $\sigma$ is the SB constant. If $T_A>T_B$ then the heat transfer is from A to B.

This is to be compared with the 1900 proof by Planck based on particle/quanta statistics typically expressed on the following form involving only one blackbody of temperature $T$:

• $Q = \sigma T^4.$       (2)

Comparing (1) and (2) we see that (1) expresses the radiative heat transfer between two bodies in resonance, while (2) is supposed to express the radiative heat transfer from one body independent of surrounding bodies and thus without resonance. 

In particular, (1) expresses that heat transfer from A requires the presence of a receptor B with lower temperature. On the other hand (2) appears to express that a body can radiate (spit out quanta/photons) without receptor, or assuming the presence of "empty space" at 0 Kelvin acting as receptor. In this case the body at higher temperature will spit most and so win the combat. 

This leads to the following questions: 

• Does radiative heat transfer from one body need a receptor at lower temperature?
• Does radiative heat transfer involve a resonance phenomenon between bodies? 

The new proof of PSB suggests that the answer is YES, while the standard proof suggests NO. What does physics and observation say? YES or No? Is radiative heat transfer carried by electromagnetic waves or particles/photons? An answer that it is both is no good. The questions concern basic physics and must be answered.

Compare with resonance between two tuning forks:

 To be compared with a particle model with both forks spitting out particles/phonons?

PS Read about Planck's struggle to prove (2) in Quantum Mechanics at the Crossroads starting with Schrödinger Against Particles and Quantum Jumps by M. Bitbol and continuing with Max Planck's Compromises on the Way to and from the Absolute by J. L. Heilbron.

Yes, it is not a good idea to resort to compromises in science, which is the essence of politics. Planck was not happy with his particle/quanta statistics and neither was Schrödinger, yet it has come to serve as a fundamental part of quantum mechanics following Born-Bohr. Real Quantum Mechanics in the spirit of Schrödinger presents a new realist deterministic approach based on waves instead of particle statistics.  

 

Discussion with Will Happer on Temperature vs Radiation Measurement

The discussion with Will Happer recorded in the previous post concerns the question what pyrgeometers and bolometers primarily measure: (a) temperature or (b) radiative heat energy flux. Will says (b) and I say (a). 

My argument is that if you look into the design of pyrgeometer, you see that it uses a thermopile, which is a device which reports a voltage which scales with the temperature difference between the two ends of the thermopile with one end by radiative equilibrium taking the same temperature as a distant source and the other end in contact with an ordinary thermometer which can be read. After calibration you can then from measured voltage and temperature read determine the temperature of the distant source. A pyrgeometer thus acts a thermometer which can read temperature at distance. This is what an infrared camera does.

A bolometer works in a similar way using a sensor with resistance scaling with temperature difference vs a thermal reservoir at constant known temperature.   

Will has another conception of pyrgeometers and bolometers:

1. Spectral intensity measurements are often expressed as equivalent temperatures. 
2. But the basic measurement is of energy fluxes which produce voltages or currents in sensor elements.

Will thus claims that the basic measurement is radiative energy flux and not temperatures. How can this be? 

Here 1 gives a clue: Will says that in some sense temperature and radiative energy flux are "equivalent". But what is this equivalence? After all, temperature is a state variable depending on the state of a system while radiative energy flux is a process variable depending on the process involved. In any case the standard procedure is to connect radiative flux $Q$ to temperature $T$ by the Planck/Stefan/Boltzmann Law for black/grey body radiation

• $Q=\epsilon\sigma T^4$      (1) 

where $\sigma$ is the Stefan-Boltzmann constant $\epsilon$ emissivity. This gives radiative flux the quality of a state variable, but this runs the risk to be misleading, since the process aspect is forgotten. A correct process version of the PSB Law reads

• $Q=\epsilon\sigma (T^4 - T_b^4)$   (2)

where $T_b$ is a background temperature like the temperature of the thermal reservoir for the bolometer. Only if $T_b=0$ does (1) give a correct connection between temperature and radiative energy flux and in addition the emissivity enters as an unknown to determine. Compare with next post.

The confusion increases by letting the pyrgeometer on its display show Downward Long Wave Radiation which is computed from the measured voltage as shown in PS3 below using (1) to express Outgoing Long Wave radiation from the instrument. A pyrgeometer thus measures temperature but reports radiative energy flux by using (1), which does not involve the process. You can thus be fooled by a pyrgeometer, which may be hard to accept if you just have bought one.   

My conclusion: 

• Pyrgeometers and bolometers do what is physically possible, namely to directly measure the temperature of a source by putting a thermometer in close or distant radiative contact with the source. 
• On the other hand, to measure radiative energy flux is very difficult since a whole process is involved with many unknowns and that is not what pyrgeometers and bolometers can do.
• See presentation at Climate Sense 2018. 

I am waiting for Will's conclusion.

PS1 Read about the Kipp and Zonen CGR3 Pyrgeometer:

• The CGR3 is a pyrgeometer, designed for meteorological measurements of downward atmospheric long wave radiation. 
• The CGR3 provides a voltage that is proportional to the net radiation in the far infrared (FIR). 
• By calculation, downward atmospheric long wave radiation is derived.

It is clearly stated that the pyrgeometer is a ghost detector serving climate alarmism measuring one thing (net temperature difference) and reporting something else (gross downward atmospheric long wave radiation), which is derived by (1). This has become so accepted, that even many skeptics believe in what the instrument display shows, although it defies scientific sense. If you have invested in a Kipp and Zonen CGR3 Pyrgeometer, you may not want to hear that you have bought a ghost detector, unless you want to send a ghost CO2 alarm…

PS2 You may compare measuring the difference between your body temperature and the surrounding room temperature, which is easy to do, with measuring how much your body is radiatively heated by the colder walls of the room, which is impossible without a ghost detector.

PS3 From the Kipp and Zonen Instruction Manual:

We see that the instrument display does not show net recording as the voltage $U_{emf}$ but instead gross $L_d$ as DWLR being computed from Formula 1 with $5.67*10^{-8}*T_b^4=\sigma T_b^4$ the incorrect gross outgoing upwelling ghost radiation from the pyrgeometer as if it was in radiative contact with outer space at 0 Kelvin. By claiming gross upwelling radiation, gross downwelling radiation is concluded, but the upwelling radiation is not real but ghost radiation. 

To well understand (not get fooled by) what an instrument display reports, it is necessary to look into the design of the instrument by reading the manual to see what is effectively measured and what is displayed. This is what I did above.