Ned Nikolov & Karl Zeller: Exact Calculations of Climate Sensitivities Reveal the True Cause of Recent Warming

Posted: May 2, 2022 by tallbloke in Analysis, atmosphere, climate, Clouds, cosmic rays, Dataset, IPCC, modelling, Natural Variation, physics, pressure, radiative theory, research, solar system dynamics
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I’m delighted Ned Nikolov and Karl Zeller have chosen the Talkshop as the venue for the publication of this new open peer review paper on climate sensitivity. Scientific advance at the cutting edge has always been the most important aim of this blog, and I think this paper truly is an advance in our understanding of the climate system and the factors which support and modulate surface temperature on Earth and other rocky planets. 

The paper is mathematically rigorous, but is also accessible to everyone, thanks to Ned and Karl’s exemplary effort to fully explain their concepts and definitions in terms which can be understood by any interested reader who has some familiarity with the climate debate. Building on the bedrock of their 2014 and 2017 papers, this new work extends the applicability and validates the postulates of those previous papers by examining the causes of variability in planetary surface temperature and incorporating the previous findings in quantifying and deriving equations to model them. They find that Earth is sensitive to changes in cloud cover, which affects the amount of solar shortwave radiation reaching the surface, but not very sensitive to changes in Total Solar Irradiance arriving at the top of the atmosphere. They also find that the sensitivity to changes in CO2 levels has been heavily overestimated by current climate models. They show that a doubling of atmospheric CO2 concentration from 280 ppm to 560 ppm will cause an undetectable global warming of 0.004K.

A PDF of the paper can be downloaded here:  ECS_Universal_Equations.

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Exact Formulas for Estimating the Equilibrium Climate Sensitivity of Rocky Planets & Moons to Total Solar Irradiance, Absorbed Shortwave Radiation, Planetary Albedo and Surface Atmospheric Pressure.
Ned Nikolov, Ph.D. and Karl Zeller, Ph.D.
April, 2022

1. Introduction

The term “Equilibrium Climate Sensitivity” (ECS) has become a synonym for the steady-state response of global surface temperature to a modeled long-wave radiative forcing caused by a doubling of atmospheric CO2 concentration with respect to an assumed pre-industrial level of 280 ppm. According to climate models based on the Greenhouse theory, an increase of atmospheric CO2 from 280 ppm to 560 ppm would produce a net radiative forcing (i.e. an atmospheric radiant-heat trapping) of 3.74 W m-2 (Gregory et al. 2004) resulting in a global surface warming between 2.5 K and 4.0 K with a central estimate of 3.0 K according to IPCC AR6 (see p. 11 in Climate Change 2021: The Physical Science Basis. Summary for Policymakers). This implies an average unit ECS of 3.0/3.74 = 0.8 K / (W m-2) with a range of 0.67 ≤ ECS ≤ 1.07 K / (W m-2). Contemporary climate science and IPCC Assessment Reports do not discuss global temperature sensitivities to changes in cloud albedo, absorbed solar radiation or total surface atmospheric pressure. Consequently, no equations have been derived/proposed thus far to calculate these sensitivities. The reason for such an omission is the implicit assumption made by IPCC based on the 19th-Century Greenhouse theory (Arrhenius 1896) that the observed warming during most of the 20th Century and especially over the past 40 years was chiefly caused by an increase of industrial CO2 emissions, which are believed to trap outgoing long-wave radiation in the Earth’s troposphere and reduce the rate of surface infrared cooling to Space.

However, a plethora of studies published during the past 15 years have shown through both satellite and surface observations that the absorption of solar radiation by the Earth-atmosphere system has increased significantly since 1982 due to a decreased cloud cover/albedo, a phenomenon often referred to as “global brightening” (e.g. Goode & Pallé 2007; Wild 2009; Herman et al. 2013; Stanhill et al. 2014; Hofer et al. 2017; Pfeifroth et al. 2018; Pokrovsky 2019;  Delgado-Bonal et al. 2020; Dübal & Vahrenholt 2021;  Yuan et al. 2021). This implies a global warming driven by a rising surface solar radiation rather than CO2.

While the CO2 “radiative forcing” is a model-generated quantity, the brightening of Earth’s surface over the past 4 decades has been inferred from actual instrumental measurements. Nevertheless, the climate sensitivity to variations of shortwave fluxes has largely been ignored by the mainstream science. An a-priori assumption has been made that the sensitivity of global temperature to any type of radiative forcing inside the system should equal the modeled ECS to CO2. In this article, we’ll show that the Earth’s ECS to shortwave solar radiation is quantitatively quite different from the hypothesized ECS to CO2. To this end, first we will derive universal analytical models for computing ECS of rocky planets and moons to changes in solar radiation, planetary albedo, and total atmospheric pressure. Secondly, we will verify the albedo-temperature model against CERES satellite measurements of Earth’s reflected shortwave radiation obtained during the past 20 years. Finally, we will apply the new analytical models to compare climate sensitivities of Earth to those of other planetary bodies in the Solar System and discuss reasons for estimated differences.

2. Derivation of Analytical Models of Equilibrium Climate Sensitivities

Analytical models are mathematical expressions with a closed form solution, which means that the solution to a differential equation describing the change of a system’s parameter is exact and can be expressed as a mathematical analytic function.

While analyzing NASA planetary data, Nikolov & Zeller (2017) made a discovery that the long-term (baseline) global surface temperature of rocky planets and moons (Tsb ,K) is mainly a function of two variables: Total Solar Irradiance (TSI) reaching the top of the atmosphere and the mean atmospheric pressure at the surface. In mathematical terms:

image002

where Tna(S) is the global average surface temperature in the absence of atmosphere (i.e. the no-atmosphere temperature), which chiefly depends on TSI (S , W m-2); and Ea(P) is the Relative Atmospheric Thermal Enhancement (RATE), a dimensionless quantity describing a form of adiabatic heating caused by the gravity-induced force of air pressure (P , Pa). Figure 1 displays the planetary bodies and their observed key parameters utilized in the Dimensional Analysis of Nikolov & Zeller (2017).

Studied_Planetary_Bodies
Figure 1. Planetary bodies in the Solar System with available high-quality observations of environmental variables used in the Dimensional Analysis of Nikolov & Zeller (2017). Note that A and Ts in this Figure correspond to 𝛼b and Tsb in the text.

Volokin & ReLlez (2014) showed that the airless global surface temperature  of a spherical body is given by the formula:

image010

where 𝛼e is the albedo of the surface regolith under airless conditions (fraction); ηe is the fraction of absorbed daytime solar radiation stored in the regolith and released as heat at night; Rc = 3.13𝑒 − 6 is the cosmic background radiation (W m-2); Rg is the average geothermal heat flux at the surface (W m-2); ε is the regolith long-wave emissivity (≈0.98); and
σ = 5.67e − 8 W m-2 K-4 is the Stephen-Boltzmann constant relating the radiative flux from a body to the 4th power of the body’s absolute temperature. Equation 2 was derived via spherical integration of the Stephen-Boltzmann radiation law.

RATE is calculated using an empirical function derived via non-linear regression analysis of data from 6 planets and moons spanning a vast range of physical conditions in the Solar System (Fig. 1), i.e.

image017

where Pr = 0.61173 kPa is a reference pressure assumed to equal the triple point of water. The purpose of using a reference pressure in Eq. 3 is to make the regression coefficients independent of pressure-measurement units. Note that the empirical coefficients in Eq. 3 differ somewhat from those published by Nikolov & Zeller (2017), because the regression analysis has been updated since the original paper using newer and better data for the baseline planetary temperatures of Venus, Earth, and Titan. Figure 2 depicts the curve described by Eq. 3.

image019
Figure 2. Graphical depiction of the Relative Atmospheric Thermal Effect (RATE), a form of pressure-induced adiabatic heating empirically described by Eq. 3.

A key new insight from the NZ model (Eq. 1) is that the climate system is not solely driven by radiation, which is a form of diabatic (external) heating, but it is also controlled by an adiabatic enhancement of the absorbed solar energy (internal heating) due to air pressure. Adiabatic heating is a standard thermodynamic phenomenon in compressible fluids such as gases. The Greenhouse theory of climate change exclusively focuses on radiative forcing and positive radiative feedbacks, and does not consider the adiabatic warming effect of atmospheric pressure on a planet’s surface.

Nikolov & Zeller (2017) demonstrated that, for bodies with tangible atmospheres, Eq. 2 can be simplified (without sacrificing numerical accuracy) by using constant generic values for 𝛼e and ηe based on NASA’s Moon data and ignoring the small energy-flux terms Rc and Rg i.e.

image021

This reduces Eq. 1 for planetary bodies with an atmosphere to the following simple expression:

image022

Equation 5 does not contain explicit terms for the absorbed solar radiation or the cloud albedo. Yet, it provides a robust mathematical basis for the derivation of exact analytical formulas to quantify planetary climate sensitivities to incoming shortwave radiation, albedo, and total surface pressure. That’s because this integral model accurately and completely describes the baseline global surface temperature of planetary bodies over a broad range of physical environments in the Solar System (see Figures 1 and 2). As such, Eq. 5 can be combined with the rules of calculus to produce closed-form solutions for various equilibrium climate sensitivities defined in terms of perturbations to the baseline global temperature.

2.1 Modeling the Sensitivity of Global Temperature to Total Solar Irradiance

The sensitivity of Tsb to TSI can be inferred from the total derivative of Tsb with respect to S, dTsb/dS. Using Eq. 5 in combination with the chain rule of calculus, we obtain:

image024

Since the mean surface atmospheric pressure (P) is a function of gravity and the mass of an atmospheric column above a unit surface area, P can be viewed as being independent of S for small variations of a planet’s orbit around the Sun such as those caused by Milankovitch cycles. This implies ∂P⁄∂S = 0, which reduces Eq. 6a to:

image026

Thus, the total derivative dTsb/dS becomes equal to the partial derivative of dTsb with respect to S , which can be obtained by differentiating Eq. 5:

Equation_7

Upon separation of variables and integrating both sides of Eq. 7, one arrives at the expression:

image030

which has the closed-form solution:

image031

In Eq. 8, ΔTsb is a change of the baseline temperature Tsb caused by a TSI perturbation Δs. Hence, the sensitivity of global temperature to TSI is proportional to the current baseline temperature of a planet and increases logarithmically with the magnitude of the TSI perturbation.

2.2 Modeling the Global Temperature Sensitivity to Absorbed Solar Radiation

The sensitivity of Tsb to absorbed solar radiation (Sa , W m-2) can be evaluated using a similar approach to the one employed in Section 2.1. Applying the chain rule of calculus to Eq. 5 yields the following expression for the total derivative dTsb/dSa :

image036

Again, since the mean atmospheric pressure P at the surface is independent of Sa for small variations of the absorbed shortwave flux caused by albedo fluctuations, we can safely assume ∂P/∂Sa = 0 , which simplifies Eq. 9a to:

image038

The partial derivative ∂Tsb/∂S was already evaluated in Eq. 7. The second partial derivative ∂Sa/∂S can be obtained from the standard formula for calculating the average absorption of shortwave radiation by a sphere:

image041

where αb is the planet’s long-term (baseline) Bond albedo defined as the phase-integrated fraction of incident solar radiation at the top of the atmosphere that is reflected back to Space, and thus lost to the climate system. Differentiating Eq. 10 with respect to S yields:

image042

Combining Equations 7, 9b and 11 produces a differential formula describing ECS to Sa :

Equation_12

Quantifying the equilibrium temperature response ΔT (K) to a finite change of the absorbed shortwave flux ΔSa (W m-2) requires a separation of variables in Eq. 12 followed by integration of both sides, i.e.:

image047

Equation 13 has the closed-form solution:

image048

Replacing Sa in Eq. 14 with its equivalent from Eq. 10 yields the final analytical formula for calculating ΔT as a function of ΔSa:

image049

In Eq. 15, ΔT is the deviation of global surface temperature from a baseline value Tsb. Similar to the TSI sensitivity, here ΔT is also proportional to Tsb and varies logarithmically with the radiation-absorption perturbation ΔSa.

2.3 Modeling the Sensitivity of Global Temperature to Planetary Albedo

Since the albedo is a key parameter determining the amount of solar radiation absorbed by a planetary body, we can use Eq. 15 as a starting point to derive a formula for the global temperature response ΔT to a finite albedo perturbation Δα . To this end, first we differentiate Eq. 10 with respect to αp:

image052

The solution to this is simply

image053

Next, we replace ΔSa in Eq. 15 with its equivalent from Eq. 17 to arrive at an analytical albedo-temperature formula:

image054

Using Eq. 18 we can now write a mathematically robust expression describing the global surface temperature of a rocky planet or moon (Ts ) as a function of 3 terms: (a) the no-atmosphere global surface temperature Tna(S) being chiefly a function of TSI; (b) the pressure-induced adiabatic atmospheric thermal enhancement Ea(P); and (c) the temperature anomaly ΔT caused by a departure of the planet’s albedo (Δα) from a baseline value αb (Eq. 18) i.e.

image056

Upon replacing the three terms in Eq. 19 with their equivalent expressions from Equations 2, 3 and 18 we arrive at a generic model describing the average global surface temperature of rocky planets and moons with atmospheres of arbitrary amount and composition:

Equation_20

Equations 18 through 20 have important new implications for the role of albedo in planetary climates that can be summarized as follows:

a) TSI and the mean atmospheric pressure at the surface determine the baseline (long-term) global surface temperature (Tsb ) of rocky planets while also giving rise to a baseline albedo . Hence, αb is a byproduct of the climate system. Being an intrinsic property of that system, αb does not affect Tsb. This conclusion follows from the 2017 Nikolov-Zeller model (Eq. 1), which accurately describes the long-term global surface temperatures of planetary bodies over a vast range of environments in the Solar System without explicitly accounting for differences in Bond albedos. Vetted NASA observations suggest that, across a broad range of physical environments, αb cannot be predicted from measured temperatures and atmospheric pressures. This fact further reinforces the notion that αb is an emergent parameter of the climate system rather than a controller of climate. Such an understanding about the physical nature of αb explains the observed stability of planetary albedos, since atmospheric pressure and TSI that give rise internally to αb tend to be stable over long periods of time.

b) If TSI and surface air pressure are constant, then the global surface temperature Ts can deviate from Tsb only if the planet’s cloud albedo is forced to depart from its baseline value. Hence, the albedo only affects a planet’s global temperature if Δα ≠ 0.0. Since Δα is much smaller than αb due to negative feedbacks operating within the climate system that constrain albedo fluctuations, the bulk of the albedo on any planet or moon with a tangible atmosphere has no impact on Ts. This implies that large positive ice-albedo feedbacks simulated by theoretical models are likely unreal, which is counterintuitive and constitutes a new finding in climate science.

For a more comprehensive discussion about the role of albedo in climate, please watch this video  presented at the 101st AMS Meeting in January of 2021.

2.4 Modeling the Global Temperature Sensitivity to Total Atmospheric Pressure

Current climate science does not recognize direct thermodynamic effects of atmospheric pressure on the global surface temperature. The “Greenhouse” theory only acknowledges the influence of pressure on temperature through the pressure broadening of gaseous infrared absorption lines. The semi-empirical model by Nikolov & Zeller (2017) is the only one that properly quantifies the Atmospheric Thermal Effect as a form of a pressure-induced adiabatic heating operating on rocky planets and moons with atmospheres. This makes the NZ model uniquely suited for evaluating the climate sensitivity to a change of total air pressure (Δp ). Since the atmospheric thermal enhancement Ea(P ) described by Eq. 3 is an explicit integral function of pressure, one does not need derivatives and the chain rule of calculus to come up with a correct climate-sensitivity model. Instead, one must simply perform a differencing of Eq. 1 with respect to pressure to calculate the climate sensitivity ΔTsb (K) to this thermodynamic forcing i.e.

image063

Note that Eq. 21 quantifies the response of the baseline temperature itself to a change of total surface pressure Δp. In contrast, formulas describing climate sensitivities to variations of albedo and the absorbed solar radiation (Equations 15 and 18) evaluate the deviation of global temperature ΔT from a baseline value Tsb. This principal difference is due to the fact that TSI and total atmospheric pressure are the variables defining a planet’s baseline temperature Tsb (Eq. 1).

3. Verification of the Albedo-Temperature Model against CERES EBAF Data

We decided to test the hypothesis that global temperature variations in recent decades were caused by changes in cloud albedo rather than atmospheric CO2 concentration. To this end, we inverted Eq. 20, which incorporates the new analytic albedo-temperature model (Eq. 18) to estimate monthly and annual changes in Earth’s albedo (Δα ) and the Reflected Solar Flux (RSF) at the top of the atmosphere from observed global near-surface temperature records provided by two official data sets: the satellite-based UAH and the surface-based HadCRUT4. Reported temperature anomalies by UAH and HadCRTU4 were converted to absolute global surface temperatures by assuming that, during the 1981 – 2010 period, the Earth’s average surface air temperature was 287.2 K (Jones & Harpham 2013). A value of 0.3 was used for the baseline albedo in Eq. 20 corresponding to a pre-industrial global baseline temperature of 286.4 K.  TSI was quantified in our model using the AcrimSat observational record. RSF was calculated from modeled Δα using Equations 10 and 17. Next, we compared the modeled dynamics of RSF to reflected shortwave radiation independently measured from orbit by the Clouds and the Earth’s Radiant Energy System (CERES) from 2001 to 2019. We utilized Edition 4.1 of the CERES Energy Balanced and Filled (EBAF) data product. If albedo anomalies (Δα ) predicted by the inverted Eq. 20 (which contain no “greenhouse-gas forcing”) using observed global surface temperatures from two independent sources agree with satellite-measured changes of reflected shortwave radiation by CERES, then our hypothesis would be considered validated.

Figure 3 shows the results from the model-data comparison using smoothed monthly data. Figure 4 illustrates comparison results based on annual data. Note that the modeled reflected solar fluxes fall within one third of the CERES calibration uncertainty range, which indicates a tight functional relationship between the planetary albedo and global surface temperature.

image064
Figure 3. Monthly dynamics of modeled reflected shortwave radiation by Earth based on Eq. 20 and near-surface global temperature records compared to observed reflected shortwave fluxes shifted 7 months forward independently measured by CERES.

image065
Figure 4. Annual dynamics of modeled reflected shortwave radiation by Earth based on Eq. 20 and near-surface global temperature records compared to observed reflected shortwave fluxes independently measured by CERES.

Changes of modeled albedo inferred from near-surface temperature records employing different measurement methods match remarkably well the interannual variation and the multi-year trend of measured reflected solar radiation by CERES. This suggests an albedo control over the global surface temperature variations since 2001. Our analysis also showed that the best model-data match is achieved when reflected CERES shortwave fluxes are shifted 7 months forward. This implies that the global surface temperature responds to changes of cloud albedo with a lag of 7 months. The presence of such a lag further strengthens the argument that observed interannual variations and the overall multidecadal trend of global temperature are indeed driven by changes in Earth’s cloud albedo rather than human CO2 emissions. Prior published research has shown that Sun’s activity likely forces changes in Earth’s cloud albedo either directly via modulation of the interplanetary electric field by solar wind (Voiculescu et al. 2013) or indirectly through the Sun’s magnetic field constraining the flux of galactic cosmic rays into the Earth’s troposphere. Cosmic rays are known to ionize air molecules and boost the production rate of cloud-condensation nuclei, thus increasing the low-level clouds (Svensmark et al. 2021). Although the exact mechanisms of cloud control by the Sun are not sufficiently understood yet to be mathematically incorporated into climate models, it is becoming increasingly clear that, on a decadal time scale, Earth’s climate is governed by the parameter Δα in Eq. 20, not anthropogenic CO2. Nevertheless, more research is needed in the area of magnetic/electric interactions between Earth and the Sun, and the effect of such interactions on cloud dynamics. In this regard, it’s important to point out that, according to recent satellite observations, the interplanetary Space is not electrically neutral as previously thought but instead is filled with plasma-enabled electric currents (a.k.a. Birkeland currents) measuring millions and billions of Amperes (see this 2018 EOS article entitled “Electric Currents in Outer Space Run the Show”). Climate models do not simulate the ionizing and electric effects of cosmic rays and the solar wind on cloud dynamics.

The high accuracy of the modeled reflected solar fluxes inferred from two independent global temperature datasets (Figures 3 and 4) validates our hypothesis that Earth’s climate of the 21st Century is most likely driven by fluctuations of cloud albedo rather than an elusive CO2 “radiative forcing” inferred from theory.

The above results also indicate that the hereto derived analytical models quantifying equilibrium climate sensitivities to variations of albedo and absorbed solar radiation are mathematically correct and physically robust. In a follow-up article soon to appear on this blog, we will apply Equations 15 and 18 to reassess the evolution of Earth’s global surface temperature over the past 60 years using a new gridded data set of measured Surface Solar Radiation (SSR) produced by Yuan et al. (2021). The article will also analyze the observed increase of SSR (global brightening) since 1982 as a driver of recent global warming.

4. Comparing Equilibrium Climate Sensitivities of Planetary Bodies Across the Solar System

The above results raise confidence in the ability of the new analytical models to correctly quantify the ECS to key forcing agents. This justifies the application of these models to compare equilibrium sensitivities of planetary climates across the Solar System. Table 1 provides such a quantitative comparison for the most studied planetary bodies: Venus, Earth, Moon, Mars, Titan and Triton.

Table 1. Equilibrium Climate Sensitivity (ECS) of planetary bodies in the Solar System to Total Solar Irradiance (TSI), absorbed solar radiation, total albedo, and surface atmospheric pressure. ECS refers to a steady-state change of the global surface temperature in response to a unit forcing.

Physical Parameter Venus Earth Moon Mars Titan Triton
Average Distance to the Sun (AU) 0.7233 1.0 1.0 1.5237 9.582 30.07
Total Solar Irradiance (S, W m-2) 2,602.1 1,361.3 1,361.3 586.4 14.8 1.5
Baseline Global Temperature (Tsb, K) 699.0 287.2 197.4 190.6 93.0 39.0
Baseline Bond Albedo (αb, faction) 0.90 0.293 0.136 0.235 0.265 0.65
Surface Atmospheric Pressure (P, kPa) 9,300 98.55 3e-13 0.6854 146.7 0.004
ECS to TSI: K / (W m-2), Eq. 8b 0.067 0.053 0.036 0.081 1.518 4.966
ECS to Absorbed Solar Radiation:
K / (W m-2), Eq. 15
2.666 0.298 0.168 0.423 7.269 20.97
ECS to Total Albedo:
K / (0.01 albedo increase), Eq. 18
-18.412 -1.023 -0.575 -0.627 -0.318 -0.283
ECS to Surface Atmos. Pressure:
K / kPa, Eq. 21
0.075 0.161 40.053 4.913 0.038 4.33

.
Climate sensitivities show a complex pattern of variation among the studied bodies due to differences in baseline surface temperatures, Bond albedos, and total atmospheric pressures. In general, the ECS to shortwave radiation increases with Tsb and αb , and decreases with TSI (S). The ECS to pressure variations is high for nearly airless bodies in relatively close proximity to the Sun such as the Moon and progressively declines in a non-linear fashion with P, approaching zero for bodies with massive atmospheres such as Venus or with sizable atmospheres but located far away from the Sun such as Titan. This is explained in part by the strongly nonlinear response of RATE to surface air pressure (see Fig. 2).

The equilibrium sensitivities of Earth’s global temperature to shortwave radiation (i.e. TSI and the absorbed solar flux) are much lower than assumed by the Greenhouse theory based on a modeled ECS to CO2. This is because climate models simulate numerous positive feedbacks, which are fictitious in nature, that amplify the initial system response to a CO2 “radiative forcing” between 2 and 4.5 times. However, as demonstrated by Nikolov & Zeller (2017), the real climate system has no measurable sensitivity to ambient CO2 due to a minute contribution of this trace gas to the total pressure of Earth’s atmosphere. Distinguishing between a theoretical (model-generated) internal forcing and a measured external climate forcing is crucial for advancing our understanding and predictive capabilities. The incoming solar radiation and its dynamic modulation by the water-cloud albedo appear to be the real forcing of Earth’s climate on decadal to centennial time scales. The relatively low ECS of Earth to TSI and absorbed solar radiation ensures a potentially greater stability of our climate compared to that of other planetary bodies such as Mars, Titan and Triton. For example, Earth’s 0.053 K / W m-2 sensitivity to TSI implies that expected variations of Sun’s luminosity and Earth’s orbit causing annual TSI fluctuations in the order of 1 – 5 W m-2 did not and will not ever have a significant impact on Earth’s climate. However, the modulation of Earth’s cloud cover affecting the planet’s absorption of solar energy forced either directly by the solar wind or indirectly by the Sun’s magnetic field through its effect on the galactic cosmic ray flux, is expected to have a sizable impact on global temperature that is 4.4 to 10 times greater than the impact of TSI fluctuations alone. This is because Earth’s ECS to absorbed solar radiation is nearly 6 times higher than the sensitivity to TSI, and the decadal variability of shortwave absorption is typically larger than TSI variability. Note in Table 1 that a 1% shift in Earth’s albedo would cause a -1 K change in the global surface temperature. To put this sensitivity into a perspective, consider that, according to the HadCRUT5 surface temperature record, 1 K is just about the entire warming experienced by Earth since 1850, i.e. over a period of 170 years.

The ECS to albedo variations might be an indicator of how strong the internal feedbacks are that maintain (support) the bulk of planetary albedos as an intrinsic property of the system. Among the bodies listed in Table 1, Venus has by far the highest climate sensitivity to albedo perturbations due to its hot surface and a strongly reflective cloud cover (see Eq. 18). This implies that the Venusian albedo is also likely to show the smallest temporal variations among the studied bodies. Earth has the second highest ECS to albedo perturbations suggesting that the albedos of Mars, Titan and Triton might be more dynamic (less stable) on decadal-to-centennial time scales compared to the Earth’s albedo.

The above estimate of Earth’s ECS to total pressure (0.161 K/kPa) can be used to calculate the response of global surface temperature to a doubling of atmospheric CO2 compared to a preindustrial level of 280 ppm. According to the Nikolov-Zeller discovery about the adiabatic nature of the atmospheric thermal effect, a change in the amount of any gas in the atmosphere (including CO2) impacts global temperature only through the contribution of such a change to total surface air pressure. In other words, what matters for the global thermal environment at the surface is the partial pressure of gases, not their infrared radiative properties. Thus, a 280 ppm increase of atmospheric CO2 implies a 0.0276 kPa increase of surface air pressure (i.e. 98.55*280/106 = 0.0276 kPa). Multiplying this perturbation by the ECS to pressure yields the true response of our planet’s global surface temperature to a CO2 doubling: 0.0276*0.161 = 0.0044 K. This amount of global warming is practically undetectable. Hence, current climate models overestimate the Earth’s global temperature sensitivity to atmospheric CO2 about 682 times or by 68,100% on average (i.e. 3.0/0.0044 = 681.8).

5. Conclusion

Derivation of exact analytical formulas for estimating the equilibrium climate sensitivities of planetary bodies to shortwave radiative forcing and surface atmospheric pressure was possible thanks to a new robust model of global surface temperature inferred from NASA planetary data by Nikolov & Zeller (2017). The model provides novel insights about the role of albedo in climate and into the physical nature of the Atmospheric Thermal Effect (currently called “greenhouse effect”) as a form of adiabatic heating caused by total pressure that is independent of atmospheric composition. The ECS Equations 8b, 15 and 18 were derived from the NZ model employing standard rules of differentiation and integration in calculus. Previous attempts to estimate ECS have been focused on the Outgoing Long-wave Radiation (OLR) as a temperature controller based on an a-priori assumption in the “greenhouse” theory that the atmosphere warms Earth by impeding the rate of surface radiative cooling to Space, a process also known as radiant-heat trapping or cooling retardation. However, the rate of cooling is never a limiting factor in the energy budget of open systems such as the atmosphere, because reducing cooling requires a form of thermal insulation that either impedes conduction/convection or reflects back thermal radiation. None of these mechanisms are operating in a free atmosphere. Since OLR is an effect (consequence) of atmospheric and surface temperatures, this infrared flux cannot affect such temperatures especially in a thermodynamic environment characterized by uninhibited energy dissipation through turbulent convection and advection. The approach of using OLR to evaluate climate sensitivity yields erroneous results also in part because it relies on fictional (non-physical) parameters such as the “effective radiating temperature” and the “effective emission altitude” (for details, see Volokin & ReLlez 2014). A study by Harde (2017) provides a recent example of employing this flawed approach and obtaining completely incorrect ECS estimates as a result. Focusing on OLR as a climate controller instead of analyzing incoming shortwave fluxes that heat the system diabatically is backward in regard to the chain of physical causality. Earth’s climate is controlled by the amount of absorbed solar energy and the adiabatic enhancement of such energy by atmospheric pressure, not by OLR. Hence, a planet’s global surface temperature is independent of the atmospheric long-wave radiative transfer and the rate of infrared cooling to Space, because these are byproducts of the climate system.

Combining the original NZ model with an analytical formula that quantifies the response of global temperature to albedo perturbations (Eq. 18) produced Eq. 20, which fully describes the global surface temperature of rocky planets and moons without recourse to a greenhouse-gas radiative forcing. The latter is a model-generated quantity based on a conjectural 19th-Century hypothesis, which is not supported by modern satellite observations. For example, the classical definition of the “greenhouse effect” as a difference of outgoing long-wave fluxes between the surface and the top of the atmosphere (Ramanathan 1989; Schmidt et al. 2010) yields physically nonsensical results over central Antarctica, where the “greenhouse effect” becomes negative (Schmithüsen et al. 2015Sejas et al. 2018). However, the actual atmospheric thermal effect over the Earth’s South Pole measured with respect to the thermal environment of the Moon’s airless South Pole is about 144 K (Fig. 5). Hence, the radiative “greenhouse effect” as currently defined has no meaningful relationship to the actual surface warming caused by the presence of an atmosphere. This is not surprising since the “greenhouse effect” was arbitrarily defined by Prof. Veerabhadran (Ram) Ramanathan (at the Scripps Institution of Oceanography, University of California, San Diego) as a radiative flux difference in the 1980s based on nothing else but his a-priori belief that the atmosphere acts as a blanket trapping heat, which is thermodynamically incorrect. An open, convective atmosphere without a lid on top cannot trap heat and does not impede cooling! Prof. Ramanathan admitted contriving his definition of the “greenhouse-effect” in a 2014 paper entitled “Climate Change and Protection of the Habitat: Empirical Evidence for the Greenhouse Effect and Global Warming“ that was published in a periodical of the Vatican City called “Complexity and Analogy in Science: Theoretical, Methodological and Epistemological Aspects“. He erroneously assumed that the difference of thermal radiative fluxes between the surface and the top of the atmosphere measures “the thickness of the greenhouse blanket”.

image066
Figure 5. The atmospheric thermal effect over Central Antarctica evaluated with respect to the airless thermal environment at the South Pole of the Moon.

The ability of Eq. 20 to accurately reproduce a 20-year trend and interannual variability of reflected solar radiation measured by CERES using observed records of near-surface global temperature as input (Figures 3 and 4) constitutes a physical proof that the recent warming was caused by a reduction of cloud albedo, not a rise of greenhouse-gas concentrations as claimed by the IPCC.

The robust derivation of Equations 8b, 15, 18 and 21 makes it meaningful to apply these models to other planetary bodies in the Solar System in order to compare changes in ECSs along a cosmic environmental gradient. Estimates shown in Table 1 indicate that Earth has a relatively low ECS to shortwave radiation compared to other bodies, which makes Earth’s climate perhaps more stable. Earth’s sensitivity to absorbed solar radiation (~0.3 K/W m-2) is 2.7 times lower than the typical modeled sensitivity to a CO2 “radiative forcing” (0.8 K/W m-2). The reality is that the Earth’s ECS to CO2 is essentially zero due to a minuscule contribution of this gas to the total atmospheric pressure on our planet. It’s also worth mentioning that Earth’s ECS to TSI is about 6 times lower than the planet’s sensitivity to absorbed solar flux. Earth has a relatively high climate sensitivity to variations of cloud albedo (-1.02 K/1% albedo change), which indicates the presence of relatively strong negative feedbacks within the system that tend to stabilize albedo fluctuations. This is good news for our global climate.

A PDF of the paper can be downloaded here:  ECS_Universal_Equations

Comments
  1. Ned Nikolov says:

    This is likely a highly technical paper for most non-scientists. So, please take your time to study it carefully in order to fully understand it, since it provides a full scientific proof that CO2 is not a driver of modern climate change.

  2. stpaulchuck says:

    our current society has fallen in love with kleptocracy, rent seeking, and other criminal behaviors for personal enrichment. What is truly egregious is that the Fourth Estate and ‘science’ have both fallen to this.

    If this was the late 50’s or so, AGW would have been argued and found totally wanting and all the charlatans of the Klimate Kaliphate would have been drummed out of scientific circles.

  3. oldbrew says:

    Just from the introduction I was reminded of this…

    Clouds Dominate CO2 as a Climate Driver Since 2000
    January 9th, 2010 by Roy W. Spencer, Ph. D.

    The main point I am making here is that, no matter whether you assume the climate system is sensitive or insensitive, our best satellite measurements suggest that the climate system is perfectly capable of causing internally-generated radiative forcing larger than the “external” forcing due to increasing atmospheric carbon dioxide concentrations. Low cloud variations are the most likely source of this internal radiative forcing.

    https://www.drroyspencer.com/2010/01/clouds-dominate-co2-as-a-climate-driver-since-2000/

  4. Phoenix44 says:

    It’s fine to demonstrate that the Earth is slightly warmer because its a bit less cloudy, but why is it a bit less cloudy?

  5. oldbrew says:

    current climate models overestimate the Earth’s global temperature sensitivity to
    atmospheric CO2 about 682 times or by 68,100% on average (i.e. 3.0/0.0044 = 681.8)!

    So much for ‘carbon footprints’.

  6. Johna says:

    Used with Joseph Postma’s spherical earth model real time parameterisations, this should prove conclusively to the politicians who are sceptical, but too afraid to speak out against the UN for fear of loosing their job, that CO2 outgassing and or intrinsic CO2 does not drive the climate on Earth. And I don’t wish to be an alarmist either, but the issue in the Ukraine could end up as a World War as UN NATO Fascist Nazi Elitist forces are intent on putting many more lives in jeopardy because of their CO2 narrative. But it has manifested in the exponential increase in the cost of living for the public to subsidies their untenable energy policies. And yes people are quite right to be angry and to fight against those who are effectively oppressing them by said CO2 narrative

  7. Schrodinger's Cat says:

    In answer to Phoenix44, less air pollution may be part of the answer. As well as being able to absorb, scatter or reflect incoming SW radiation, some dispersed species may contribute to conventional cloud seeding.

  8. Ned Nikolov says:

    What Roy Spencer wrote in 2010 about the role of clouds is conceptually correct. However, he seems to have abandoned this line of thought recently. Roy has increasingly been pushing the false “greenhouse” theory and insisting that adding CO2 to the atmosphere via industrial emissions would measurably warm the planet.

    We should be asking, why is he dismissing the evidence accumulated since 2010 that observed changes in absorbed solar radiation by Earth is sufficient to explain the entire recent climate change and that there is no evidence for the so-called CO2 “radiative forcing”?

  9. oldbrew says:

    I hear what you say Ned. In an earlier (2008) blog post RS wrote:

    Here I present new evidence that most of the warming could be the result of a natural cycle in cloud cover forced by a well-known mode of natural climate variability: the Pacific Decadal Oscillation (PDO).
    . . .
    As Joe D’Aleo, Don Easterbrook, and others have pointed out for years, the Pacific Decadal Oscillation (PDO) has experienced phase shifts that have coincidently been associated with the major periods of warming and cooling in the 20th Century.

    https://www.drroyspencer.com/research-articles/global-warming-as-a-natural-response/

  10. Ned Nikolov says:

    Indeed, Oldbrew!

    I’d only add to this 2008 statement by Roy Spencer that oceanic oscillations such as PDO, AMO, ENSO etc. are all caused by changes of regional cloud cover/albedo forced by Sun’s activity in conjunction with the Earth’s electromagnetic field rather than these oscillations causing cloud cover variations… So, these are not really phenomena of internal natural climate variability. They are externally forced!

  11. oldbrew says:

    Yes, it’s the original cause that really counts. Not found in most climate models it seems.

    Do they have a definition of ‘natural’ as in climate variation?

  12. tallbloke says:

    ENSO ~ 3 El Nino peaks per solar cycle – the big ones occur soon after solar minimum.
    AMO ~ 6 solar cycles in 2 groups of three. 1st group has two +ve and one -ve; 2nd group the opposite.

  13. […] and Tallbloke are continuing (I am grateful for them, too). They have arrived at the subtle question of Earth’s […]

  14. Ned Nikolov says:

    People on Twitter seem to like our paper. Thanks for tweeting it, Roger:

    Now that Elon Musk is in charge of Twitter, I filed an appeal yesterday to unlock my account, which was suspended last summer after posting some info about Bill Gates and his murky involvement in the COVID pandemic mismanagement… I hope Twitter is now becoming a free-speech platform again.

  15. tallbloke says:

    It would be great to have you back on twitter Ned. You have new fans.

  16. Phoenix44 says:

    Oldbrew, “original cause” is I think an error. Complex non-linear systems have no starting point and everything affects everything. Every change is a change to a change, not to a stasis or fixed position. Its why modelling the starting point of the model is impossible, because its all “change”. Many modelers and activists believe in a “stable” climate which CO2 perturbs but that seems very unlikely as climate is never the same (in total) twice.

  17. oldbrew says:

    This might be of interest, re. planetary bodies.

    MAY 3, 2022
    Experiments measure freezing point of extraterrestrial oceans to aid search for life

    “Our results show that the cold, salty, high-pressure liquids found in the deep ocean of other planets’ moons can remain liquid to much cooler temperature than they would at lower pressures. This extends the range of possible habitats on icy moons, and will allow us to pinpoint where we should look for biosignatures, or signs of life.”

    https://phys.org/news/2022-05-extraterrestrial-oceans-aid-life.html

  18. tallbloke says:

    Phoenix44: Complex non-linear systems have no starting point and everything affects everything.

    True if cloud cover is an internal variable. Not true if it’s driven by an external variable such as solar wind or cosmic ray flux.

  19. Phoenix44 says:

    Tallbloke, it’s still true if it’s external. The external forcing exists but changes. There are no stable external forcings, they constantly change for all sorts of reasons and the system is constantly changed by those changes. And the change on the “climate” caused by a change in the external forcing is not linear. Something like solar wind is not on-off, it is variable. Yes, it has no feedback from our climate so in that sense you are right, but its effects depend on the climate’s state.

  20. tallbloke says:

    It was the “everything affects everything” I was taking issue with. External variables such as solar wind and cosmic ray flux may well be constantly changing, but they’re not being changed by Earth’s cloud fraction.

  21. bobweber says:

    “The incoming solar radiation and its dynamic modulation by the water-cloud albedo appear to be the real forcing of Earth’s climate on decadal to centennial time scales.”

    I think this is true; however your theory/formulae also need to be field-tested with ongoing data.

    The reliance for cloud generation on cosmic rays in this work is a non-starter, so this paper should be revised or retracted. The Svensmark cloud-climate theory is empirically wrong.

    During the time of highest cosmic rays, the world experienced record sunshine from fewer clouds:

  22. SamH says:

    Hello N&Z, your integral 8b is mistaken, as are similar ones that follow. You cannot treat Tsb as a constant on one side of the equation, while it is a variable of integration on the other side.

    Equation 8b is obtained by taking the derivative of equation 5, then integrating the result. Because those operations are inverses, this should just give equation 5 back.

  23. Ned Nikolov says:

    SamH,

    Equation 8b and the ones that follow are all correct, because these are the result of integration of partial derivatives to Eq. 5. Each partial derivative is different, hence the resulting integral solutions are different as well… Please consult a calculus textbook!

  24. Ned Nikolov says:

    Bobweber,

    The sensitivity equations presented in our paper do not contain any information about the cause of cloud-albedo variations. They only quantify the impact of such variations on the global surface temperature. There is experimental and satellite-measured evidence that cosmic rays affect the rate of production of cloud condensation nuclei. There is also evidence that solar wind directly affects cloud formation. But the mechanisms are in general poorly understood at present. This is why we stated that more research is needed to pinpoint the exact forcing of cloud albedo variations…

    What our analysis has unequivocally demonstrated is that observed global temperature changes in recent decades are caused by cloud-albedo fluctuations, not atmospheric CO2!

  25. SamH says:

    Hi Dr. Nikolov,

    Please check again, the integral is incorrect. You are beginning the partial derivative of Tsb with respect to S, then integrating with respect to S. These are inverse operations.

  26. Ned Nikolov says:

    SamH,

    The purpose of integrating the partial derivative with respect to S is to find an analytical expression quantifying the change of temperature deltaT in response to a change of TSI deltaS. This integration is different from Eq. 5, which describes the total absolute temperaures Tsb as a function of total TSI. I hope you understand the distinction now… If not, please explain how the integral in Eq. 8b is incorrect?

    Note that Eq. 8b yields ZERO, if deltaS = 0

  27. tallbloke says:

    Ned, CO2-control knob theorists often claim reduced cloud cover is a positive feedback to warming caused by CO2. How do you repond to papers such as this one?

    https://www.pnas.org/doi/10.1073/pnas.2026290118

  28. SamH says:

    Dr. Nikolov, the problem with the integration is that Tsb was treated as a constant inside an integral over S, but this is not correct because Tsb and S vary together.

    The expression for Delta Tsb can be read off from equation 5:

    DeltaTsb = Tsb(S0+DeltaS) – Tsb(S0)

    = 32.44 Ea(P) [(S0+ DeltaS)^.25 – S0^.25]

    You could also write this as

    DeltaTsb = Tsb0 {[(S0+DeltaS)/S0]^.25 – 1}

  29. oldbrew says:

    N&Z get a mention here…

    Why are we destroying fossil fuels and our modern societies? Part 2
    By Terigi Ciccone, Dr. Jay Lehr | May 2nd, 2022

    https://www.cfact.org/2022/05/02/if-co2-does-not-contribute-to-global-warming-why-are-we-destroying-fossil-fuels-and-our-modern-societies-part-2/

    They also propose a link between El Niño/La Niña and volcanism…

    For many years it was accepted orthodoxy that these El Niño/La Niña events were due to climatic circulation patterns in the ocean and atmosphere. However, this has been dispelled by evidence that the El Niño and La Niña are one continuous geologic event in recent years. Meaning, they are caused by increased or decreased tectonic/volcanic activities in the deep ocean floors where the crust is the thinnest, [H][h] and humans have little knowledge of when, where, and how powerful [I][i]. We know this is new to most readers, but you can take it to the bank and cash it.

    [H] link — https://www.earth.columbia.edu/articles/view/3231
    ‘Seafloor Volcano Pulses May Alter Climate’

  30. tallbloke says:

    Well, maybe, but I doubt it. The regular occurence of the biggest El Ninos soon after solar minimum is too obvious to be ignored. I suppose it’s possible that low solar activity has a geomagnetic effect that increases tectonic activity in the ocean deeps, but I think Ned is right that ENSO is primarily cloud driven. That’s because the 1983 eruption of El Chichon chucked enough sulphur dioxide into the upper atmosphere to significantly reduce solar surface radiation, inducing El Nino. It reduced the size of the 1988 solar minimum el Nino, because it had already reduced the amount of energy stored in the pacific warm pool below surface.

    That indicates low activity at solar minimum allows more cloud seeding by the cosmic ray flux, also inducing the emission of energy from the ocean. Bob Weber’s comment above is noted, but there may be other things affecting multidecadal trends. The decadal situation with the link between solar cycles, CRF, cloud cover and ENSO seems clear to me.

  31. bobweber says:

    Ned, cloud nucleation experiments aren’t real clouds in the sky. Nature has provided as I showed a clear refutation of cosmic ray theory in practice. You also don’t need to go looking for cloud generation in the solar wind data either. You’re wasting your time with Svensmark’s theory.

    We can see from the low-level vs high-level tropical clouds that they are inversely and seasonally related, with a slight decline since the 1980-90s:

    Altogether tropical clouds were higher during the larger solar cycles #21 & #22, declining into #23:

    While climate4you has a plug for Svensmark’s theory too, I still don’t believe it. Global cloud cover peaked during after the 1983 El Nino, and thereafter declined after the SC22 peak into the solar minimum, and rose again with rising SC #23 solar activity.

    My recent work with TEMIS Ozone/UVI of nearly 250 stations indicates cloud cover is directly tied to ENSO extremes, and a solar cycle influence above 95 SN that drives above-average cloud cover.

    Which nixes your premise that fewer clouds are the sole reason for post-2000 warming, not TSI.

    The reduction in clouds during the ‘pause’ came about from lower solar activity, afterward SST was warmed by above 95 SN (higher TSI) in SC #24. SC #24 TSI drove SSTs upwards until 2016 when TSI fell below my decadal sun-ocean warming threshold. This had nothing to do with albedo.

    I offer as evidence for TSI warming the ocean during SC #24 the following published Nature image with my SORCE (v17) TSI overlay. The southern ocean was particularly sensitive to sunspot cycle #24 TSI changes because those TSI spikes during those solar maximum years happened near perihelion, when the sub-solar point is over the southern ocean, driving ocean absorption.

    Furthermore, UAH, HadSST3, and ERSSTv5 all indicate SST cooling from 2016 into the present, and with cloud cover diminishing from the 2015-16 El Nino (see earlier TEMIS plots), all of which contradicts your theory that fewer clouds would cause an increase in SST, albeit land temperatures were indeed high from fewer clouds.

    btw I predicted ahead of time both the high TSI SST warming into 2016 and cooling thereafter from lower TSI, and clouds were not part of my prediction system, because the state of cloud fraction changes are caused by TSI changes

    This is part of what I mean by field testing. I like your derivations Ned but your cloud connections need an upgrade.

  32. Phil Salmon says:

    Thanks N & Z!
    Needs an abstract.

    Since OLR is an effect (consequence) of atmospheric and surface temperatures, this infrared flux cannot affect such temperatures especially in a thermodynamic environment characterized by uninhibited energy dissipation through turbulent convection and advection.

    At the heart of many corrupt and fraudulent arguments is an inversion of cause and effect. I think you’re correct that this is the inversion at the core of the CO2 warming story. IR fluxes are, indeed, EFFECTS, NOT CAUSES, of atmospheric temperature.

    Atmospheres are not warm because of emitted IR.
    They emit IR because they’re warm!

  33. tallbloke says:

    I agree Phil. As we were taught at school back in my youth:
    “Everything radiates according to its temperature.”
    Not the other way round.

  34. Ned Nikolov says:

    Dear SamH (@SamH May 4, 2022 at 4:19 pm) ,

    I now see, where your confusion comes from! Your difference equation

    DeltaTsb = Tsb(S+DeltaS) – Tsb(S) = 32.44 Ea(P) [(S+ DeltaS)^.25 – S^.25]

    is correct. However, you did not realize that it produces exactly the same numerical result us our Eq. 8b for an equivalent input. That’s because these are identical equations written in different forms! For example, let’s take Earth as an illustration. Using

    S = 1361.3 W m-2 and P = 98.55 kPa

    we obtain from Eq. 5:

    Tsb = 32.44*Ea(P)*S^.025 = 285.747 K

    Using your difference equation, we get for the temperature sensitivity (assuming DeltaS = 1 W m-2):

    DeltaTsb = 32.44 Ea(P) [(S+ DeltaS)^.25 – S^.25] = 0.052 K (W m-2)

    Using our Eq. 8b, we get:

    DeltaTsb = 0.25*Tsb*ln(1 + DeltaS/S) = 0.25*285.747*ln(1 + 1/1361.3) = 0.052 K (W m-2)

    So, Eq. 8b is the integral form of your difference equation. The advantage of using Eq. 8b over your equation is that Eq. 8b describes DeltaTsb as an explicit function of Tsb. Because of this, Eq. 8b is also applicable to planetary bodies without an atmosphere, while your difference equation is strictly only applicable to bodies with tangible atmospheres. So, Eq. 8b is more general in that sense in addition to being more elegant and cleaner as a math expression.

    I hope this helps…

  35. Ned Nikolov says:

    Hi Phil Salmon,

    I intentionally omitted the Abstract, so that people have the intellectual incentive to read the whole paper… I agree with rest of your comment.

  36. Ned Nikolov says:

    tallbloke @ May 4, 2022 at 4:09 pm

    Yes, I’m aware of the doubletalk used by mainstream climate science with respect to the undeniable fact that cloud-cover has decreased over recent decades. Casting cloud-cover changes as a “feedback” to global temperature rather than acknowledging it as a primary driver of recent warming is a way to keep the physically false “greenhouse” theory afloat while avoiding the embarrassment that climate science has been spending public funds studying the wrong topic of CO2 “radiative forcing” for 40 years!

  37. SamH says:

    Dear Dr. Nikolov,

    Our two equations are indeed numerically close, but they are not quite the same. Mine is correct, and yours is the result of an improperly performed integral. Although for practical purposes the difference is small, I thought you might want to ensure the math in your paper is correct.

    My expression is:

    DeltaTsb = Tsb0 {[(S0+DeltaS)/S0]^.25 – 1}

    While yours is:

    DeltaTsb = 0.25 Tsb0 ln(1 + DeltaS/S0)

    Let us call DeltaS/S0 = x. Then aside from the shared factor of Tsb0, our formulas are (1+x)^.25 – 1, versus .25 ln(1+x). You can see the two expressions plotted at the following link, where there is clearly a difference, if a small one: https://www.wolframalpha.com/input?i=plot+.25*ln%281%2Bx%29%2C+%281%2Bx%29%5E.25-1

    I would say my expression is just as convenient and elegant as yours, but it is more correct.

  38. Ned Nikolov says:

    SamH,

    Let’s not argue about the obvious: Eq. 8b is the correct integral expression describing the response of global planetary temperature (deltaT) to a TSI perturbation (deltaS). The calculus leading to Eq. 8b is straightforward and undoubtful. That’s it!

  39. SamH says:

    Ned, I’m afraid you’re wrong. I would urge you to look carefully one more time. It is no problem to make a basic math mistake like this, but it is not a good look to allow it into your final paper.

    My formula was derived without any calculus at all, it is really quite simple.

  40. dodgy geezer says:

    What is the point of this paper? No one will read it. Global Warming Mania has gone beyond proof – you could show clear cooling and we would still have to sacrifice our energy, our infrastructure, our society and our lives to the great Climate Change God.. ..

  41. tallbloke says:

    geezer; the point is to correct the science. How long it takes after that to get the science establishment to gloss over the gross errors of the last 40 years and switch to the correction is another fight for another day. Chin up and keep telling ’em.

  42. Ned Nikolov says:

    SamH,

    Your equation:

    DeltaTsb = Tsb {[(S + DeltaS)/S]^.25 – 1}

    is simply wrong, because it does not follow from proper calculus, and cannot be derived from the simple difference expression corresponding to Eq. 5:

    DeltaTsb = Tsb(S+DeltaS) – Tsb(S) = 32.44 Ea(P) [(S+ DeltaS)^.25 – S^.25]

    You just made up an equation that produces DeltaTsb = 0, if DeltaS = 0. But that’s not how this works!

  43. tallbloke says:

    SamH, thanks for your input here. This is what open peer review is all about: in-depth discussion of the finer points. Resolution can be reached without rancour when both parties are striving to understand the other’s point of view.

  44. Ned Nikolov says:

    I’m sorry, but SamH apparently does not understand calculus and has trouble deriving one equation from another…

  45. SamH says:

    @tallbloke, thanks, I’m glad to have the opportunity for open discussion of scientific works on this website.

    @ Dr. Nikolov, I stand by my equation being the correct one, but there is not much more to say that has not already been said, so I will leave it be. I have some questions about more important aspects of the paper that I will try to write up in the next couple of days, I hope you get a chance to answer them. Thanks.

  46. Ned Nikolov says:

    Thank you, SamH.

    I’ll try to answer your more substantial questions when you post them here…

  47. bewhitebarry says:

    This would appear to confirm the work of Jiki Kauppenin & Malen of Turku University in Finland and a team at Kobe university in Japan.
    How long can they go being ignored.

  48. oldbrew says:

    However, the actual atmospheric thermal effect over the Earth’s South Pole measured
    with respect to the thermal environment of the Moon’s airless South Pole is about 144 K (Fig. 5).

    Quite a big number! I suspect Arrhenius didn’t know about that.

  49. Ned Nikolov says:

    Yes, Arrhenius did not have a clue about the atmospheric thermal effect at the Poles, since he had no idea how cold the Moon Poles were…

  50. oldbrew says:

    Too late for Arrhenius of course, but I’m reminded of this…

  51. Ned Nikolov says:

    I think today’s mainstream climate science follows the rule:

    When the facts change, we adjust the data, so we don’t have to change our mind.

    🙂

  52. tallbloke says:

    “an increase of 0.5%–0.7% in solar radiation was noted over the southern half of the country. In central Pakistan, the cloud cover decreased by 3%–5% with a consequent increase of 0.9°C in temperature. ”

    https://jaimejessop.substack.com/p/omg-15-billion-people-experiencing

  53. oldbrew says:

    Guardian: ‘North-west and central India experienced the hottest April in 122 years’ – Tallbloke’s link.

    So it was just as hot there 122 years ago. Was that a human-induced climate crisis too?

  54. bobweber says:

    @ tallbloke, solar minimum-induced La Nina caused SST cooling and higher land temperatures.

    From Jaime Jessop’s article,

    “Pakistan experienced a heatwave in late April. It’s cooled off a bit now, but due to return in May. Northern India is also experiencing heatwave conditions.”

    The heatwave is due to strong persistent and early summer-like sunshine with high UVI anomalies, extreme sunshine from cloudlessness that started in Kashmir in 2020 from the La Nina. Northern India had large, enduring UVI anomalies for most of March and April, with May starting out strong.

    Although there isn’t any ESA TEMIS ozone/uvi data for Pakistan, I’m sure they are like India and the MidEast, which showed some similar overall patterns in the TEMIS data since 2020. The USA also had early 2022 spring high UVI anomalies, driving some record high temps and wildfires.

  55. Another issue is that the carbon hockey stick is not driven by fossil fuel. It is in fact driven by soil damage tied to tilling and harvesting. This reduces soil water retention to boot, which results in more water vapor. The result is desertification, which is observable and actually driving climate change.

    You can read about this in my book, which is publicly available on my Substack:

    http://ddebernardy.substack.com/p/a-natural-language?s=w

    (I’m in the process of rearranging it as multiple posts to make it easier to share. The above url will be the summary and the table of content when I am done.)

    Kindly help get the word out by sharing the link with people who you know. In particular climate scientists, climate activists, and environmental journalists.

  56. bobweber says:

    Another example of the lack of connection between cosmic rays and clouds: two polar TEMIS ozone anomalies (cloud proxy) near 70N and S, respectively, show for the NH (Barrow, AK, USA) high ozone anomalies throughout the period starting with the 2010 El Nino and lasting until the 2016 El Nino warming subsided, going into negative integrated anomalies through the solar minimum when cosmic rays are highest. This is just the opposite of what Svensmark’s theory predicts.

    Secondly, the southern polar station at Maitri, Antarctica shows an increase in integrated ozone starting with the 2010 El Nino, increasing after SN>95 in 2011/12, persisting through the solar max-driven 2016 El Nino, and lasting through to the solar minimum reinforced by the high Nino-like 2018 tropical warmth until 2020, just after the solar minimum when ozone anomalies became more negative as the La Nina began. Nothing about this cloud history was initiated by cosmic rays. Clouds aren’t an independent variable, but they sure aren’t dependent on cosmic rays for nucleation.

  57. Ned Nikolov, Ph.D. says:

    BobWeber,

    No one is arguing that cosmic rays are the only factor affecting cloud cover. I’ve come to believe for a number of years now that cloud cover is controlled by many processes most of them electric and magnetic in nature. Solar wind and Sun’s open magnetic fields, as well as Moon’s gravitational influence on the jet streams appear to be factors impacting cloud formation as well.

    My article above makes the following points:

    1. Changes of cloud albedo precede variations of surface temperature and are the driver of Earth’s climatic fluctuations on decadal to centennial time scales;

    2. Changes of cloud albedo are likely externally forced by cosmic factors interacting with Earth’s magnetic field.

    That’s it!

  58. tallbloke says:

    Excellent contributions, thanks everyone.
    What the above indicates to me is that the investigation of the causes of natural variation have been neglected by mainstream climate science, in its dogged (and dogmatic) pursuit of CO2 as the primary (sole) climate control knob.

    As Ned said in the paper:
    “more research is needed in the area of magnetic/electric interactions between Earth and the Sun, and the effect of such interactions on cloud dynamics. In this regard, it’s important to point out that, according to recent satellite observations, the interplanetary Space is not electrically neutral as previously thought but instead is filled with plasma-enabled electric currents”

    Has anyone here got around to reading Brian Tinsley’s papers on the ‘global electrical circuit’?

  59. […] Study: Exact Calculations of Climate Sensitivities Reveal the True Cause of Recent Warming […]

  60. oldbrew says:

    ‘plasma-enabled electric currents’

    Like the solar wind…
    https://en.wikipedia.org/wiki/Solar_wind

  61. oldbrew says:

    Universe Today is pushing the Venus runaway greenhouse theory again…

    Volcanoes May Have Killed Venus with a Runaway Greenhouse, and Almost the Earth Too
    MAY 8, 2022

    What turned Venus into hell? It could have simply been a steadily-warming Sun, but new research suggests that Volcanoes may have played a role in creating a runaway greenhouse effect. And the same history of active Volcanism almost killed the Earth, too.

    https://www.universetoday.com/155746/volcanoes-may-have-killed-venus-with-a-runaway-greenhouse-and-almost-the-earth-too/

  62. Ned Nikolov, Ph.D. says:

    What turned Venus into a hot hell, was the built-up of huge atmospheric pressure (93 bar), which adiabatically warmed the surface to 699 K. The massive Venusian atmosphere might well be a result of large-scale volcanism (there are thousands of active volcanoes on Venus today!), but it does not matter what kind of gases are being emitted by volcanoes as long as the overall pressure rises…

  63. […] (LinkkiPDF) kiviplaneettojen tasapainotilaisesta ilmastoherkkyydestä ECS. Jutun löysin alunperin Tallbloke-blogista (Linkki). Tutkielman tulos on ilmastotieteen valtavirran tulkinnan räjäyttävä: [The reality is […]

  64. Dave Evenden, PhD says:

    @Dr. Nikolov, @SamH, (and @tallbloke)

    I was intrigued by the mathematical solution quandary, so here’s my ha’pence worth. I use online and other solvers to explore and to do the heavy lifting. I am not a mathematician btw but a former chartered engineer, and now post-doc academic researcher.

    Anyhow, here’s what I get. I hope the links work (if not the raw latex-like equation can be pasted into the grey box):

    https://www.symbolab.com/solver/calculus-calculator/%20%5Cint_%7BT_%7Bsb%7D%7D%5E%7BT_%7Bsb%7D%2B%5CDelta%20T_%7Bsb%7D%7DdT%20%3D0.25T_%7Bsb%7D%5Cint_%7BS%7D%5E%7BS%2B%5CDelta%20%20S%7D%5Cfrac%7BdS%7D%7BS%7D?or=input

    or (paste & click go):
    \:\int _{T_{sb}}^{T_{sb}+\Delta \:T_{sb}}dT\:=0.25T_{sb}\int _S^{S+\Delta \:\:S}\frac{dS}{S}

    Simplifying…(trivial standard log identity)

    https://www.symbolab.com/solver/algebra-calculator/simplify%20%5Cleft%5B0.25T_%7Bsb%7D%5Cleft(%5Cln%5Cleft%7Cs%2B%CE%94s%5Cright%7C-%5Cln%5Cleft%7Cs%5Cright%7C%5Cright)%5Cright%5D?or=input

    or:
    simplify\:\left[0.25T_{sb}\left(\ln \left|s+Δs\right|-\ln \left|s\right|\right)\right]

    The upshot is a result identical to eqn 8a. If there’s another solution approach, it might be useful to explore and present this in these terms.

    Maybe worth a side chat somehow, rather than the main thread.

  65. Ned Nikolov, Ph.D. says:

    Dave,

    The calculus used in our paper is VERY basic and straightforward! There cannot any doubt about the solutions I presented, because these follows from the Table of Standard Integrals. See Eq. 2 on this page:

    http://integral-table.com/

    One does not need a numerical solver for integrals that have a closed-form solution like the equations in the paper. Our math only looks complicated to people, who have no understanding of calculus, which is unfortunately the vast majority of the public. It’s sad to see the bitter fruits of a substandard modern education system that fails to teach students the basics of higher math. Luckily, I belong to an older generation that was taught these things…

  66. tallbloke says:

    Ned, you don’t look all that old. Neither am I. Fighting the CO2 fanatics has given me grey hair though.

  67. Joe says:

    @Ned
    Would it be correct to argue that atmospheric content is negligible as to the mass of the atmosphere?

  68. […] Study: Exact Calculations of Climate Sensitivities Reveal the True Cause of Recent Warming […]

  69. Ned Nikolov, Ph.D. says:

    Joe, that is an ill-posed question, because composition and mass are totally different types of parameters. Any atmosphere has a content of gases and these gases may have any kind of mass… So, your question does not make much sense!

  70. Joe says:

    @Ned
    “Joe, that is an ill-posed question, because composition and mass are totally different types of parameters. Any atmosphere has a content of gases and these gases may have any kind of mass… So, your question does not make much sense!”

    As you see I’m not a scientist and I clearly asked the wrong question. I’m trying to understand if the content in the atmosphere has any influence on surface temps? Your unified theory of climate seems to argue no. But I forever hear increased content will increase mass causes a rise in surface equilibrium.

  71. Linnea says:

    @Joe & Ned

    I think Joe is confused as to the point of this work. Which is fine, these are not easy things for those not versed in science. Ned’s findings show the added CO2 content (or its mass) didn’t increase the surface temp.

    Thr added content (say CO2 for example) is negligible, because solar radiation absorption by the atmosphere increased significantly at the same time as the purported surface temp increase. Especially since 1982, which has been driven by a change in albedo not content. Two things determine the surface equilibrium as stated in this paper, TSI and the mean atmospheric pressure at earth’s surface.

  72. Linnea says:

    @Ned & Old Brew
    “What turned Venus into a hot hell, was the built-up of huge atmospheric pressure (93 bar), which adiabatically warmed the surface to 699 K.”

    Venus to me is the opposite of what is argued as to the GHE. Especially a runaway GHE which is utter nonsense. The Venusian atmospheric pressure of 9000 kPa if placed on earth would get the same surface temp irrelevant of which gases existed within.

    On earth, temperatures increase by about 80C going from 20 to 100 kPa. Which means at 9000 kPa we would see 20C + ln(9000/(100-20)) *80C = 400C as earth’s surface temp.

  73. Bazz says:

    The effect of the clouds is due to several cycles working together.
    The suns radiance cycle
    The Milanovitch cycles, earth’s orbit, the rotation of the earth ellipse, the tilt of the earth axis and a intensity cycle of sunspots.
    That upsets the earths magnetic field which diverts away cosmic rays.
    The means less clouds are formed and so the earth gets warmer.
    There are many know warming cycles, most recent the Roman Warming peaked around 100 BC and then got cold around 400 AD, then got warm for the Medieval Warm period around 900 ad, then got cold around 18th century
    and then got warm again around 2000 !
    These cycles known to Egyptians and Minoans.
    Nothing new to see here.

  74. Ned Nikolov, Ph.D. says:

    Bazz,

    Milankovitch cycles operate on time scales of 10s to 100s of thousands of years. Cloud cover/albedo changes operate on decadal to centennial time scale… So, Milankovitch cycles have NOTHING to do with cloud-albedo changes!

    Also, an objective analysis of available data shows that Milankovitch cycles have NO relationship even to long-term variations of Earth’s global temperature such as those observed during the Ice Ages of the past 800,000 years. See this blog post:

    Dispelling the Milankovitch Myth“: https://tallbloke.wordpress.com/2022/01/03/ned-nikolov-dispelling-the-milankovitch-myth/

  75. Ned Nikolov, Ph.D. says:

    Linnea,

    Our model described by Eq. 1 in the above blog article can accurately predict the warming that Earth would experience, if the surface air pressure were to increase to 93 bar (9,300 kPa) matching the current pressure on Venus… Earth’s global surface temperature will increase to 311.3 C, not 400 C as you’ve calculated. That’s because you’ve used an incorrect equation! 🙂

  76. Linnea says:

    @Ned

    “Earth’s global surface temperature will increase to 311.3 C, not 400 C as you’ve calculated.”

    Ah looking again at equation 1 got it!! I see my error thank you Ned! I bet I got closer to the answer than most could these days 🤣

    As to the high CO2 content in the the Venusian atmosphere, my theory has been it’s the result of atmospheric pressure. Agree, volcanic activity certainly aided the high percentage of CO2. But is it not possible a great deal of the CO2 was encapsulated in the surface and released as energy increased. Much like the eastern Antarctic ice core studies which showed increased energy freed encapsulated CO2 from the surface. Thus temp leads CO2 on Venus as much as it did on earth. Which would argue CO2 is predominantly irrelevant to the surface equilibrium on Venus.

  77. Bazz says:

    Not sure who I am replying to here; Yes I can see your point, I have read J Kauppenin’s paper and with the other cyclic influences he included Milanovitch. All the maths is well beyond me. Of interest just watched video about the collapse of the Maya civilisation in the cold period after the Roman warming. The video blamed drought (climate change ?) Tks for comment

  78. bobweber says:

    @Ned regarding clouds: “most of them electric and magnetic in nature.”

    You are fooling yourself BIGTIME Ned. I am fully aware of the Forbush decrease cloud study, but you should already know they hardly happen enough to count for weather, lest climate change.

    My wordpress avatar is my app image of 5-minute solar wind data, ie, the electric, magnetic, and electromagnetic data you claim is a factor. In fact I own three domain with those names, with ‘weather’ attached, the only person in the world who does, so just maybe Ned, after 8 plus years of practical experience and continuous observations I just might know something you don’t.

    What I do know Ned is you don’t know what you’re talking about when you invoke Svensmark’s theory and the solar wind wrt clouds. It’s bad science, bad theory. Give it up, you’re being stupid about it. If you can’t integrate what I’ve told you already about why his theory fails, then you sir are headed for failure. …tough love…

  79. pochas94 says:

    @bobweber:

    Let me risk an oversimplified pronouncement: Ultraviolet rays from the sun’s corona cause climate change, which follows solar activity, with a time lag of about 10 years, acting via ozone production, which causes changes in the adiabatic temperature profile and both atmospheric and oceanic circulation patterns. Any arguments?

  80. Ned Nikolov, Ph.D. says:

    BobWeber,

    I get your point, and I certainly know about “bad science”, since there is no shortage of it in modern climate science.

    Please explain what controls trends of cloud albedo that last for decades? Do you think that such long-term trends are externally forced, or do you believe that they are somehow a result of “internal climate variability” as claimed by IPCC?

    If you think it’s “internal variability”, then why do cloud-albedo changes precede global temperature changes anywhere from 7 months to 4 years?

  81. Ned Nikolov, Ph.D. says:

    One additional comment regarding the electric/magnetic forcing of clouds. There are a series of papers by Arthur Viterito published over the past 6 years that show a strong correlation between global temperature, sub-oceanic seismic activity, and the speed of movement of the North magnetic (Dip) Pole from 1979 to the present. See for example this 2017 paper:

    Shifting Plates, Shifting Poles, Shifting Paradigms
    https://www.omicsonline.org/open-access/shifting-plates-shifting-poles-shifting-paradigms-2573-458X-1000130.php?aid=90547

    Viterito’s most recent paper is from 2022: https://juniperpublishers.com/ijesnr/IJESNR.MS.ID.556271.php

    I do not agree with Viterito’s hypothesis that all these correlations were caused by geothermal heat released from the bottom of the World’s oceans. I think these correlations point to the Sun’s active role in controlling (influencing) global seismicity, cloud cover, and the migration of the North Dip Pole electromagnetically. One thing can be concluded with certainty based on Viterito’s research, however: Human carbon emissions were NOT the driver of Earth’s climate over the past 40 years as claimed by IPCC!

  82. Paul Cottingham says:

    My precise calibration of the Greenhouse effect of 133 Kelvin by molar mass for the Earth is. Nitrogen imparts 100.6 Kelvin, Oxygen imparts 30.3 Kelvin, Argon is third accounting for 1.7 Kelvin, and Water vapour is fourth accounting for 0.3 Kelvin. Carbon Dioxide only produces 0.0532 Kelvin by molar mass at 400ppm. So would not 280ppm impart 0.03724 or 0.04 Kelvin?

  83. oldbrew says:

    Is this on the N&Z radar?

    Earth’s Albedo Puzzle – A Question Of Balance
    Tuesday 26th April 2022 | Dr David Whitehouse, Science Editor

    You only have to look at a globe of the Earth to realise what a lop-sided planet we live on, most of the land is in the Northern hemisphere (NH), most of the ocean in the Southern hemisphere (SH). Since we were able to make space-based measurements of the Earth’s reflectance – its albedo – that dichotomy has become a puzzle as for some reason, despite their differences, the Northern and Southern hemispheres reflect the same amount of sunlight to within observational uncertainties!

    https://www.netzerowatch.com/earths-albedo-puzzle-a-question-of-balance/

  84. Ned Nikolov, Ph.D. says:

    @Paul Cottingham

    The Earth’s “greenhouse effect” is about 90 K, not 133 K. We’ve shown this in an extensive paper in 2014:
    On the average temperature of airless spherical bodies and the magnitude of Earth’s atmospheric thermal effect“: https://springerplus.springeropen.com/articles/10.1186/2193-1801-3-723

  85. Ned Nikolov, Ph.D. says:

    @oldbrew

    Yes, the amazing hemispherical symmetry of Earth’s total albedo (despite 11% difference in surface albedos between NH and SH) has been documented by at least 2 independent studies now:

    – Stephens et al. (2015) The albedo of Earth. Rev. Geophys.,53,141–163, doi:10.1002/2014RG000449. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2014RG000449

    – Datseris G and Stevens B. (2021) Earth’s albedo and its symmetry. AGU Advances, 2, e2021AV000440. DOI: 10.1029/2021AV000440. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2021AV000440

    And here is a 2021 paper based on flawed climate models that tries to explain away the observed hemispherical albedo symmetry as a “transient phenomenon” caused mainly by human aerosol emissions in the Northern Hemisphere: https://www.essoar.org/doi/abs/10.1002/essoar.10511017.1

    The model-driven junk climate science never misses an opportunity to downplay and/or distort phenomena that do not fit into the CO2-climate paradigm… 🙂

  86. oldbrew says:

    Thanks Ned. I note the Essoar climate modelling paper has zero citations.

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