Ned Nikolov and Karl Zeller have sparked a storm of controversy in climate science by publishing a study describing a mathematical model that accurately predicts Global Mean Annual near-surface temperature for all rocky planets in the Solar System, including the Earth. The controversy: the model is accurate irrespective of atmospheric composition, undermining the widely accepted theory of Climate Change driven by the Greenhouse Effect. A further study by Robert Ian Holmes details a comprehensive climate model based on adiabatic compression of gases by gravity, in which changes in the concentration of greenhouse gases have negligible influence on atmospheric warming.

The paper at the centre of the controversy, titled Emergent model for predicting the average surface temperature of rocky planets with diverse atmospheres, was published in Advances in Space Research under pseudonyms (actual names of the authors spelled backwards). The true identities of the authors were soon discovered and the journal withdrew the paper for reasons “not related to the scientific merit of the study”. The pair explained that the pseudonyms were used only to avoid prejudiced peer review on account of the authors’ earlier research. An expanded version of the paper was subsequently published in Environment Pollution and Climate Change under the title: New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary Temperature Model.

Zeller & Nikolov identified a range of parameters that could influence the energy balance of a planet (including the atmospheric density of greenhouse gasses) and conducted parametric curve-fitting for different combinations of parameters. The study specifically excluded “non-monotonic functions such as polynomials because of their ability to accurately fit almost any dataset given a sufficiently large number of regression coefficients while at the same time showing poor predictive skills beyond the calibration data range.” All parameters specific to greenhouse gases resulted in lower accuracy than the model based on atmospheric pressure alone, by a wide margin. The surprising result: partial density of greenhouse gasses had negligible influence on atmospheric temperature and the planetary thermal enhancement. The authors conclude that the Global Mean Annual near-surface temperature must be a function of solar irradiance and atmospheric pressure under a constant gravitational gradient, independent of atmospheric composition and the concentration of greenhouse gasses. If this is correct then Pleistocene glacial cycles (ice ages) must correspond to anomalies in Earth’s atmospheric pressure. Since there are no known geo-chemical proxies for historical surface air-pressure changes, this hypothesis cannot be confirmed at this time. Another source of doubt is the set of dimensionless variables used for curve-fitting as it may have failed to capture the critical variables in their critical mathematical arrangement. For example, the dimensionless parameter for the Greenhouse Effect [P^3/(ρ.S^2)] includes solar irradiance on the inverse side of the partial pressure of greenhouse gases, one essentially opposing the effect of the other, and yet both these variables could be positively related to atmospheric temperature. A more representative account of the effect of greenhouse gases could be captured with a dimensionless parameter consisting of (Mgh/Mp).(S/Sr), where Mgh is the total mass of greenhouse gases in the atmosphere, Mp is the mass of the planet, S is solar irradiance per unit area, Sr is some reference value of irradiance. Alternatively, the mass ratio could be replaced by a density ratio, based on average near-surface greenhouse gas density over some reference density value, as I have done below. To obtain the independent dimensionless parameter (X-axis) I divide greenhouse gas density by a reference value of 1.0 [kg/m^3] and solar irradiance by 1.0 [W/m^2]. Changing the magnitude of the reference did not affect the results.

The main difference between my analysis and that of Zeller-Nikolov is that I have used a different estimate for Mars’ average surface temperature, at 240K instead of the unusually low 190K assumed by the authors. I was also able to obtain an excellent fit using only a 2-parameter exponential curve-format instead of the 4-parameter format used by the authors which was criticised by some for overfitting. Incidentally, reproducing the regression analysis of Zeller-Nikolov Model 12 (based only on atmospheric pressure) with a different estimate for Mars’ surface temperature has resulted in a poor fit. The results are very sensitive to the estimated temperatures and this is likely to be the primary source of error.

Apart from objections about methodology and the lack of a comprehensive climate theory to ground the results, the pressure-driven thermal enhancement model faces another serious objection. If greenhouse gases pose no special resistance to the outgoing thermal radiation then, for a constant planetary albedo (thermal reflectivity), any thermal enhancement at the surface would result in higher emission of thermal radiation (black-body radiation), progressively cooling the planet as if it had no atmosphere at all. Holmes (2018) argues that atmospheric thermal enhancement via adiabatic compression of gases by gravity is perfectly consistent with the idea that greenhouse gases absorb and re-emit long-wavelength thermal radiation, precisely as claimed by proponents of the Greenhouse Effect theory, but the excess greenhouse gases do not result in anomalous warming. Any addition of heat causes expansion of the atmosphere because the atmosphere is not constrained in volume. Consequently, any increase in temperature would cause upward gas expansion and thus “increase potential energy at the expense of kinetic energy – so cooling the air again.”

According to the classical Greenhouse Effect theory, Earth’s surface absorbs shortwave solar radiation and re-emits a significant portion of the absorbed energy as longwave thermal radiation. While this radiation does not affect certain gasses, like oxygen and nitrogen, which are transparent to infrared radiation, it is absorbed by greenhouse gasses (water vapor, carbon dioxide, methane, nitrous oxide and ozone) and then re-emitted in all directions, some of it being emitted back towards Earth’s surface, thereby trapping heat within the lower atmosphere. This is known as radiative forcing; the degree of warming is said to depend on the concentration of greenhouse gases in the atmosphere. Holmes’ adiabatic auto-compression model of thermal enhancement is consistent with the premise that longwave thermal radiation is absorbed by greenhouse gases at the same rate as it is re-emitted when in a state of thermodynamic equilibrium; Schwarzschild’s equation of radiative transfer applies only under this condition. The critical implication of the auto-compression model is that if the amount of energy absorbed by greenhouse molecules exceeds the equilibrium value, the affected gas gains marginally higher kinetic energy and therefore marginally higher pressure, and converts any excess kinetic energy (heat) into potential energy via upward expansion; this is (allegedly) possible because the atmosphere is not constrained in volume. The Ideal Gas Law dictates that unconstrained marginal volumetric expansion of a thermodynamically isolated system causes marginal cooling, rapidly re-establishing equilibrium. If this is true then doubling the concentration of greenhouse gases at a constant pressure, density and thermal irradiation doubles the amount of kinetic energy absorbed by the gases, but the excess kinetic energy now forces the system to expand against gravity towards a new thermodynamic equilibrium. In the process, half of the absorbed kinetic energy is converted to potential energy while the atmospheric temperature and the overall rate at which thermal energy is re-emitted by the greenhouse gases do not change. This would nonetheless show up in the spectrum of longwave radiation emitted into space, as depressions in the outgoing black-body radiation profile of the planet, precisely as if the bands of radiation specific to greenhouse gases were reflected back to Earth. The fact that the intensity of some bands of radiation is reduced by doubling the concentration of greenhouse gases does not mean that the associated energy is converted to heat, as proponents of the classical Greenhouse Effect theory assume. This would happen only to a mixture of gases in a sealed container, but since the atmosphere is unconstrained in volume the ‘missing’ energy does the work of atmospheric expansion instead.

Clearly, the same bands of thermal radiation must be emphasised in the downgoing frequency profile at the surface of the planet, since only greenhouse gases can emit longwave thermal radiation, but they do so ONLY insofar as the atmosphere is in thermodynamic equilibrium. Any radiative forcing beyond the equilibrium (all else being equal) simply causes atmospheric expansion, according to Holmes. On this account, the only capacity for thermal forcing by greenhouse gases is related to their slightly higher molar mass than oxygen and nitrogen. This accounts for only 3.7% of the expected thermal forcing under the classical Greenhouse Effect model (Holmes 2018, 115).

I have serious doubts about this part of the argument. When heat is slowly added to greenhouse gases as thermal radiation, isobaric expansion follows and equilibrium is re-established, but not ALL of the added heat is used to do the expansion work. In order to maintain constant pressure only about 30% of the excess heat is required to do the work while the remaining 70% manifests as an increase in the temperature. So in that regard the official story prevails. There may be a lot of negative feedback in the system, as per the discussion below, but greenhouse gases evidently do contribute some thermal forcing, and not necessarily just radiative forcing because convection and conduction may also play a role. The question therefore is not if but how much forcing occurs on account of greenhouse gasses.

The phenomenon of atmospheric expansion and contraction is uncontroversial. It is known that “contracted thermosphere, where many satellites, including the International Space Station, operate… reduces atmospheric drag on satellites.”(Phys.org) In the thermosphere, the phenomenon of atmospheric contraction is caused by cooling associated with the build up of CO2: “CO 2 is the primary radiative cooling agent and fundamentally affects the energy balance and temperature of this high-altitude atmospheric layer.” (Emmert et al. 2012) Currently there is a “global increase in CO x (CO 2 and CO, combined) concentrations of 23.5±6.3 ppm per decade at an altitude of 101 km, about 10 ppm per decade faster than predicted by an upper atmospheric model.” This suggests that a substantial portion of CO2 migrates to altitudes where, according to Emmert (2012), “thermal energy is transferred via collisions from other atmospheric constituents to CO2, which then emits the energy as heat that escapes to outer space.” Continuous migration of CO2 from the troposphere to the upper atmosphere where it acts as a cooling agent suggests that until full mixing occurs the upper atmosphere will be subject to ongoing cooling. The net heat flux for the planet is thus likely negative at this point in time.

The upper atmospheric cooling in turn extracts heat from the troposphere via convection/mixing.This contributes to the negative thermal feedback associated with the increased concentration of greenhouse gases in the atmosphere. Another possible source of negative thermal feedback is the positive relationship between albedo and cloud-cover (Stephens et al. 2015). Warming causes increased evaporation which may in turn result in increased cloud-cover, causing the atmosphere to reflect a greater fraction of the incoming solar radiation back into space, therefore atmospheric cooling. “The effect is not small, a mere 1% change in albedo being a greater forcing than all the anthropogenic forcing claimed by the IPCC from 1750 to date.”

Despite almost univocal acceptance of the Greenhouse Effect as the primary driver of climate change, there is no empirical evidence that greenhouse gasses contribute to net [global] changes in the atmospheric temperature. There is only one study (Feldman et al. 2015) which has quantified correlation between changes in downward heat flux (0.2W/m²) and changes in CO2 concentration between 2000-2010, at two measurement sites. The study nonetheless ignores the already discussed stratospheric cooling associated with CO2, which in turn cools the lower troposphere via convection. Looking at radiative forcing alone is by no means indicative of an increase in the global temperature as it does not capture all the energy inputs and outputs affected by CO2.

As discussed above, the adiabatic auto-compression model described by Holmes is not negated by the radiative properties of greenhouse gases, but differs only in the effect this has on the atmosphere. In the classical Greenhouse Effect model, increased radiative forcing causes Global Mean Annual near-surface temperature to rise; in the adiabatic auto-compression model it causes incremental expansion of the atmosphere and no net warming. But the adiabatic auto-compression model fails to explain how 70% of thermal energy which is not needed to re-establish pressure-equilibrium does not immediately cause the local temperature to rise. Similarly, Zeller & Nikolov may have failed to consider the most relevant combination of terms in the right mathematical relation. A good fit to empirical data based only on pressure and arbitrary parameters does not tell us much without showing that other variables do not matter, a something that the authors did not do.

The overall conclusion is that changes in atmospheric temperature are determined primarily by solar irradiance, gravity, molecular mass and the concentration of greenhouse gases. The authors of both papers fail to convince that the official story is false, but they could still be close to the truth if there is a sufficient amount of negative feedback in the system. Taking account of CO2-induced stratospheric cooling could be crucial to accurate modelling of Earth’s energy ballance. Negative feedback to anomalous warming via the greenhouse effect has not been adequately quantified, nor is there any credible evidence that the present climate is anywhere near a point where it could become unstable (switch from negative to positive feedback). While intensification of the greenhouse effect could put some stress on the ecosystem and create new challenges for certain communities, the hypothetical threat of a sudden or irreversible ‘climate catastrophe’ in foreseeable future, short of an external planetary disturbance, is currently not supported by evidence.

Possibly the strongest evidence against anthropogenic global warming comes from the following climate sensitivity study by Humlum et al. (2013): The phase relation between atmospheric carbon dioxide and global temperature. The authors demonstrate, by analysing the official data-sets, that the seasonal variations in the rate of change in the atmospheric concentration of CO2 have no measurable effect on the rate of change in the mean global temperature; a result which is radically inconsistent with the athropogenic warming model. Another way, if CO2 is the dominant cause of tropospheric warming then adding CO2 faster should cause temperature to rise faster, but this second-order correlation is not there. The three critical responses to this paper misrepresent the claims made therein and fail to convince why the results should not count as a refutation of the popular hypothesis that CO2 is the “control knob” of Earth’s climate.

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