Physical Constraints on the Climate Sensitivity

Guest essay by George White

For matter that’s absorbing and emitting energy, the emissions consequential to its temperature can be calculated exactly using the Stefan-Boltzmann Law,

1) P = εσT⁴

where P is the emissions in W/m², T is the temperature of the emitting matter in degrees Kelvin, σ is the Stefan-Boltzmann constant whose value is about 5.67E-8 W/m² per K⁴ and ε is the emissivity which is 1 for an ideal black body radiator and somewhere between 0 and 1 for a non ideal system also called a gray body. Wikipedia defines a Stefan-Boltzmann gray body as one “that does not absorb all incident radiation” although it doesn’t specify what happens to the unabsorbed energy which must either be reflected, passed through or do work other than heating the matter. This is a myopic view since the Stefan-Boltzmann Law is equally valid for quantifying a generalized gray body radiator whose source temperature is T and whose emissions are attenuated by an equivalent emissivity.

To conceptualize a gray body radiator, refer to Figure 1 which shows an ideal black body radiator whose emissions pass through a gray body filter where the emissions of the system are observed at the output of the filter. If T is the temperature of the black body, it’s also the temperature of the input to the gray body, thus Equation 1 still applies per Wikipedia’s over-constrained definition of a gray body. The emissivity then becomes the ratio between the energy flux on either side of the gray body filter. To be consistent with the Wikipedia definition, the path of the energy not being absorbed is omitted.

A key result is that for a system of radiating matter whose sole source of energy is that stored as its temperature, the only possible way to affect the relationship between its temperature and emissions is by varying ε since the exponent in T⁴ and σ are properties of immutable first principles physics and ε is the only free variable.

The units of emissions are Watt/meter² and one Watt is one Joule per second. The climate system is linear to Joules meaning that if 1 Joule of photons arrives, 1 Joule of photons must leave and that each Joule of input contributes equally to the work done to sustain the average temperature independent of the frequency of the photons carrying that energy. This property of superposition in the energy domain is an important, unavoidable consequence of Conservation of Energy and often ignored.

The steady state condition for matter that’s both absorbing and emitting energy is that it must be receiving enough input energy to offset the emissions consequential to its temperature. If more arrives than is emitted, the temperature increases until the two are in balance. If less arrives, the temperature decreases until the input and output are again balanced. If the input goes to zero, T will decay to zero.

Since 1 calorie (4.18 Joules) increases the temperature of 1 gram of water by 1C, temperature is a linear metric of stored energy, however; owing to the T⁴ dependence of emissions, it’s a very non linear metric of radiated energy so while each degree of warmth requires the same incremental amount of stored energy, it requires an exponentially increasing incoming energy flux to keep from cooling.

The equilibrium climate sensitivity factor (hereafter called the sensitivity) is defined by the IPCC as the long term incremental increase in T given a 1 W/m² increase in input, where incremental input is called forcing. This can be calculated for emitting matter in LTE by differentiating the Stefan-Boltzmann Law with respect to T and inverting the result. The value of dT/dP has the required units of degrees K per W/m² and is the slope of the Stefan-Boltzmann relationship as a function of temperature given as,

2) dT/dP = (4εσT³)^-1

A black body is nearly an exact model for the Moon. If P is the average energy flux density received from the Sun after reflection, the average temperature, T, and the sensitivity, dT/dP can be calculated exactly. If regions of the surface are analyzed independently, the average T and sensitivity for each region can be precisely determined. Due to the non linearity, it’s incorrect to sum up and average all the T’s for each region of the surface, but the power emitted by each region can be summed, averaged and converted into an equivalent average temperature by applying the Stefan-Boltzmann Law in reverse. Knowing the heat capacity per m² of the surface, the dynamic response of the surface to the rising and setting Sun can also be calculated all of which was confirmed by equipment delivered to the Moon decades ago and more recently by the Lunar Reconnaissance Orbiter. Since the lunar surface in equilibrium with the Sun emits 1 W/m² of emissions per W/m² of power it receives, its surface power gain is 1.0. In an analytical sense, the surface power gain and surface sensitivity quantify the same thing, except for the units, where the power gain is dimensionless and independent of temperature, while the sensitivity as defined by the IPCC has a T^-3 dependency and which is incorrectly considered to be approximately temperature independent.

A gray body emitter is one where the power emitted is less than would be expected for a black body at the same temperature. This is the only possibility since the emissivity can’t be greater than 1 without a source of power beyond the energy stored by the heated matter. The only place for the thermal energy to go, if not emitted, is back to the source and it’s this return of energy that manifests a temperature greater than the observable emissions suggest. The attenuation in output emissions may be spectrally uniform, spectrally specific or a combination of both and the equivalent emissivity is a scalar coefficient that embodies all possible attenuation components. Figure 2 illustrates how this is applied to Earth, where A represents the fraction of surface emissions absorbed by the atmosphere, (1 – A) is the fraction that passes through and the geometrical considerations for the difference between the area across which power is received by the atmosphere and the area across which power is emitted are accounted for. This leads to an emissivity for the gray body atmosphere of A and an effective emissivity for the system of (1 – A/2).

The average temperature of the Earth’s emitting surface at the bottom of the atmosphere is about 287K, has an emissivity very close to 1 and emits about 385 W/m² per Equation 1. After accounting for reflection by the surface and clouds, the Earth receives about 240 W/m² from the Sun, thus each W/m² of input contributes equally to produce 1.6 W/m² of surface emissions for a surface power gain of 1.6.

Two influences turn 240 W/m² of solar input into 385 W/m² of surface output. First is the effect of GHG’s which provides spectrally specific attenuation and second is the effect of the water in clouds which provides spectrally uniform attenuation. Both warm the surface by absorbing some fraction of surface emissions and after some delay, recycling about half of the energy back to the surface. Clouds also manifest a conditional cooling effect by increasing reflection unless the surface is covered in ice and snow when increasing clouds have only a warming influence.

Consider that if 290 W/m² of the 385 W/m² emitted by the surface is absorbed by atmospheric GHG’s and clouds (A ~ 0.75), the remaining 95 W/m² passes directly into space. Atmospheric GHG’s and clouds absorb energy from the surface, while geometric considerations require the atmosphere to emit energy out to space and back to the surface in roughly equal proportions. Half of 290 W/m² is 145 W/m² which when added to the 95 W/m² passed through the atmosphere exactly offsets the 240 W/m² arriving from the Sun. When the remaining 145 W/m² is added to the 240 W/m² coming from the Sun, the total is 385 W/m² exactly offsetting the 385 W/m² emitted by the surface. If the atmosphere absorbed more than 290 W/m², more than half of the absorbed energy would need to exit to space while less than half will be returned to the surface. If the atmosphere absorbed less, more than half must be returned to the surface and less would be sent into space. Given the geometric considerations of a gray body atmosphere and the measured effective emissivity of the system, the testable average fraction of surface emissions absorbed, A, can be predicted as,

3) A = 2(1 – ε)

Non radiant energy entering and leaving the atmosphere is not explicitly accounted for by the analysis, nor should it be, since only radiant energy transported by photons is relevant to the radiant balance and the corresponding sensitivity. Energy transported by matter includes convection and latent heat where the matter transporting energy can only be returned to the surface, primarily by weather. Whatever influences these have on the system are already accounted for by the LTE surface temperatures, thus their associated energies have a zero sum influence on the surface radiant emissions corresponding to its average temperature. Trenberth’s energy balance lumps the return of non radiant energy as part of the ‘back radiation’ term, which is technically incorrect since energy transported by matter is not radiation. To the extent that latent heat energy entering the atmosphere is radiated by clouds, less of the surface emissions absorbed by clouds must be emitted for balance. In LTE, clouds are both absorbing and emitting energy in equal amounts, thus any latent heat emitted into space is transient and will be offset by more surface energy being absorbed by atmospheric water.

The Earth can be accurately modeled as a black body surface with a gray body atmosphere, whose combination is a gray body emitter whose temperature is that of the surface and whose emissions are that of the planet. To complete the model, the required emissivity is about 0.62 which is the reciprocal of the surface power gain of 1.6 discussed earlier. Note that both values are dimensionless ratios with units of W/m² per W/m². Figure 3 demonstrates the predictive power of the simplest gray body model of the planet relative to satellite data.

Figure 3

Each little red dot is the average monthly emissions of the planet plotted against the average monthly surface temperature for each 2.5 degree slice of latitude. The larger dots are the averages for each slice across 3 decades of measurements. The data comes from the ISCCP cloud data set provided by GISS, although the output power had to be reconstructed from radiative transfer model driven by surface and cloud temperatures, cloud opacity and GHG concentrations, all of which were supplied variables. The green line is the Stefan-Boltzmann gray body model with an emissivity of 0.62 plotted to the same scale as the data. Even when compared against short term monthly averages, the data closely corresponds to the model. An even closer match to the data arises when the minor second order dependencies of the emissivity on temperature are accounted for,. The biggest of these is a small decrease in emissivity as temperatures increase above about 273K (0C). This is the result of water vapor becoming important and the lack of surface ice above 0C. Modifying the effective emissivity is exactly what changing CO₂ concentrations would do, except to a much lesser extent, and the 3.7 W/m² of forcing said to arise from doubling CO₂ is the solar forcing equivalent to a slight decrease in emissivity keeping solar forcing constant.

Near the equator, the emissivity increases with temperature in one hemisphere with an offsetting decrease in the other. The origin of this is uncertain but it may be an anomaly that has to do with the normalization applied to use 1 AU solar data which can also explain some other minor anomalous differences seen between hemispheres in the ISCCP data, but that otherwise average out globally.

When calculating sensitivities using Equation 2, the result for the gray body model of the Earth is about 0.3K per W/m² while that for an ideal black body (ε = 1) at the surface temperature would be about 0.19K per W/m², both of which are illustrated in Figure 3. Modeling the planet as an ideal black body emitting 240 W/m² results in an equivalent temperature of 255K and a sensitivity of about 0.27K per W/m² which is the slope of the black curve and slightly less than the equivalent gray body sensitivity represented as a green line on the black curve.

This establishes theoretical possibilities for the planet’s sensitivity somewhere between 0.19K and 0.3K per W/m² for a thermodynamic model of the planet that conforms to the requirements of the Stefan-Boltzmann Law. It’s important to recognize that the Stefan-Boltzmann Law is an uncontroversial and immutable law of physics, derivable from first principles, quantifies how matter emits energy, has been settled science for more than a century and has been experimentally validated innumerable times.

A problem arises with the stated sensitivity of 0.8C +/- 0.4C per W/m², where even the so called high confidence lower limit of 0.4C per W/m² is larger than any of the theoretical values. Figure 3 shows this as a blue line drawn to the same scale as the measured (red dots) and modeled (green line) data.

One rationalization arises by inferring a sensitivity from measurements of adjusted and homogenized surface temperature data, extrapolating a linear trend and considering that all change has been due to CO₂ emissions. It’s clear that the temperature has increased since the end of the Little Ice Age, which coincidently was concurrent with increasing CO₂ arising from the Industrial Revolution, and that this warming has been a little more than 1 degree C, for an average rate of about 0.5C per century. Much of this increase happened prior to the beginning the 20’th century and since then, the temperature has been fluctuating up and down and as recently as the 1970’s, many considered global cooling to be an imminent threat. Since the start of the 21’st century, the average temperature of the planet has remaining relatively constant, except for short term variability due to natural cycles like the PDO.

A serious problem is the assumption that all change is due to CO₂ emissions when the ice core records show that change of this magnitude is quite normal and was so long before man harnessed fire when humanities primary influences on atmospheric CO₂ was to breath and to decompose. The hypothesis that CO₂ drives temperature arose as a knee jerk reaction to the Vostok ice cores which indicated a correlation between temperature and CO₂ levels. While such a correlation is undeniable, newer, higher resolution data from the DomeC cores confirms an earlier temporal analysis of the Vostok data that showed how CO₂ concentrations follow temperature changes by centuries and not the other way around as initially presumed. The most likely hypothesis explaining centuries of delay is biology where as the biosphere slowly adapts to warmer (colder) temperatures as more (less) land is suitable for biomass and the steady state CO₂ concentrations will need to be more (less) in order to support a larger (smaller) biomass. The response is slow because it takes a while for natural sources of CO₂ to arise and be accumulated by the biosphere. The variability of CO₂ in the ice cores is really just a proxy for the size of the global biomass which happens to be temperature dependent.

The IPCC asserts that doubling CO₂ is equivalent to 3.7 W/m² of incremental, post albedo solar power and will result in a surface temperature increase of 3C based on a sensitivity of 0.8C per W/m². An inconsistency arises because if the surface temperature increases by 3C, its emissions increase by more than 16 W/m² so 3.7 W/m² must be amplified by more than a factor of 4, rather than the factor of 1.6 measured for solar forcing. The explanation put forth is that the gain of 1.6 (equivalent to a sensitivity of about 0.3C per W/m²) is before feedback and that positive feedback amplifies this up to about 4.3 (0.8C per W/m²). This makes no sense whatsoever since the measured value of 1.6 W/m² of surface emissions per W/m² of solar input is a long term average and must already account for the net effects from all feedback like effects, positive, negative, known and unknown.

Another of the many problems with the feedback hypothesis is that the mapping to the feedback model used by climate science does not conform to two important assumptions that are crucial to Bode’s linear feedback amplifier analysis referenced to support the model. First is that the input and output must be linearly related to each other, while the forcing power input and temperature change output of the climate feedback model are not owing to the T⁴ relationship between the required input flux and temperature. The second is that Bode’s feedback model assumes an internal and infinite source of Joules powers the gain. The presumption that the Sun is this source is incorrect for if it was, the output power could never exceed the power supply and the surface power gain will never be more than 1 W/m² of output per W/m² of input which would limit the sensitivity to be less than 0.2C per W/m².

Finally, much of the support for a high sensitivity comes from models. But as has been shown here, a simple gray body model predicts a much lower sensitivity and is based on nothing but the assumption that first principles physics must apply, moreover; there are no tuneable coefficients yet this model matches measurements far better than any other. The complex General Circulation Models used to predict weather are the foundation for models used to predict climate change. They do have physics within them, but also have many buried assumptions, knobs and dials that can be used to curve fit the model to arbitrary behavior. The knobs and dials are tweaked to match some short term trend, assuming it’s the result of CO₂ emissions, and then extrapolated based on continuing a linear trend. The problem is that there as so many degrees of freedom in the model, it can be tuned to fit anything while remaining horribly deficient at both hindcasting and forecasting.

The results of this analysis explains the source of climate science skepticism, which is that IPCC driven climate science has no answer to the following question:

What law(s) of physics can explain how to override the requirements of the Stefan-Boltzmann Law as it applies to the sensitivity of matter absorbing and emitting energy, while also explaining why the data shows a nearly exact conformance to this law?

References

1) IPCC reports, definition of forcing, AR5, figure 8.1, AR5 Glossary, ‘climate sensitivity parameter’

2) Kevin E. Trenberth, John T. Fasullo, and Jeffrey Kiehl, 2009: Earth’s Global Energy Budget. Bull. Amer. Meteor. Soc., 90, 311–323.

3) Bode H, Network Analysis and Feedback Amplifier Design assumption of external power supply and linearity: first 2 paragraphs of the book

4) Manfred Mudelsee, The phase relations among atmospheric CO content, temperature and global ice volume over the past 420 ka, Quaternary Science Reviews 20 (2001) 583-589

5) Jouzel, J., et al. 2007: EPICA Dome C Ice Core 800KYr Deuterium Data and Temperature Estimates.

6) ISCCP Cloud Data Products: Rossow, W.B., and Schiffer, R.A., 1999: Advances in Understanding Clouds from ISCCP. Bull. Amer. Meteor. Soc., 80, 2261-2288.

7) “Diviner Lunar radiometer Experiment” UCLA, August, 2009