Saturday, August 6, 2016

Climate Change Drivers

Summary
Thermalization and the complete dominance of water vapor in reverse-thermalization explain why atmospheric carbon dioxide (CO2) has no significant effect on climate. Reported average global temperature (AGT) since before 1900 is accurately (98% match with measured trend) explained by a combination of ocean cycles (41.7%), sunspot number anomaly time-integral (38.0%) and increased atmospheric water vapor (20.3%). 

Introduction (rev 5/8/17, 10/18/17)
The only way that energy can significantly leave earth is by thermal radiation. Only solid or liquid bodies and greenhouse gases (ghg) can absorb/emit in the wavelength range of terrestrial radiation. Ghg absorb/emit only at specific wavelengths which are characteristic for each molecule specie. In the range of terrestrial temperatures, non-ghg must transfer energy to ghg (or liquid or solid bodies) for this energy to be radiated. Note: The expression ‘greenhouse gas’ is somewhat misleading (greenhouses actually work primarily by suppressing convection). A more correct understanding is that so-called ghg can absorb/emit radiation in the wavelength range of significant infrared radiation associated with earth temperatures.

The word ‘trend’ is used here for temperatures in two different contexts. To differentiate, α-trend is an approximation of the net of ocean surface temperature oscillations after averaging-out the year-to-year fluctuations in reported average global temperatures. The term β-trend applies to the slower average energy change of the planet which is associated with change to the average temperature of the bulk volume of the material (mostly ocean water) involved.

Some ocean cycles have been named according to the particular area of the oceans where they occur. Names such as PDO (Pacific Decadal Oscillation), ENSO (el Nino Southern Oscillation), and AMO (Atlantic Multi-decadal Oscillation) might be familiar. They report the temperature of the water near the surface. The average temperature of the bulk water that is participating in these oscillations cannot significantly change so quickly because of high effective thermal capacitance [1].

This high thermal capacitance absolutely prohibits the rapid (year-to-year) AGT fluctuations which have been reported, from being a result of any credible forcing. According to one assessment [1], the time constant is about 5 years. A likely explanation for much of the reported year-to-year fluctuations is that they are stochastic phenomena in the overall process that has been used to determine AGT. Volcanic activity is occasionally also a temporary contributor but typical effects are made insignificant by 5-yr smoothing. A simple calculation shows the standard deviation of the reported annual average measurements to be about ±0.09 K with respect to the trend.


The temperature fluctuations of the bulk volume near the surface of the planet (equivalent to top 110 meters of ocean) are more closely represented by the fluctuations in the trend. The trend is a better indicator of the change in global energy; which is the difference between energy received and energy radiated.

The kinetic theory of gases, thermalization, Maxwell-Boltzmann distribution of energy among atmospheric gas molecules, some thermodynamics, the absorb/emit bands of water vapor and other ghg, and the rudiments of quantum mechanics provide a rational explanation of what happens with terrestrial thermal radiation.

Some of the mistakes made by the consensus and also other relevant issues are discussed at Ref [29].

In this document, all gas molecules which are IR active at earth temperatures are ghg molecules. This includes water vapor molecules.

Refutation of significant influence from CO2 (rev 4/21/17)
What is meant by the statement that carbon dioxide (CO2) is a greenhouse gas (ghg)? If by that is meant CO2 absorbs electromagnetic radiation with a wave length of 15 microns (including +/- a micron or so due to quantum mechanics), well, that was demonstrated in the lab a long time ago and remains true. But if by that is meant CO2 significantly contributes to global warming, there is multiple compelling evidence (most identified earlier [2] ) that CO2 has no significant effect on climate:
1. In the late Ordovician Period, the planet plunged into and warmed up from the Andean/Saharan ice age, all at about 10 times the current CO2 level [3].
2. Over the Phanerozoic eon (last 542 million years) there is no correlation between CO2 level and AGT [3, 4].
3. During the last and previous glaciations AGT trend changed directions before CO2 trend [2].
4. Since AGT has been directly and accurately measured world wide (about 1895), AGT has exhibited up and down trends while CO2 trend has been only up. [2]
5. Since about 2001, the measured atmospheric CO2 trend has continued to rise while the AGT trend has been essentially flat. [21, 13]
6. Analysis of CO2 and Temperature data 2002-2008 shows a close correlation between dCO2/dT and lower tropospheric temperature. This demonstrates that CO2 follows temperature and not the reverse. [30]

Thermalization refutes CO2 influence on climate. (rev 10/21/16, 4/21/17, 8/6/17, 9/25/17)
At a scale of the size of atoms, the atmosphere consists of gas molecules with empty space between them. Activity of the gas molecules determines what can be measured as temperature and pressure. Imagery of the activity of the molecules making up the atmosphere is helpful. Wikipedia, in the article on kinetic theory of gases, has a pretty good 2-D animation of the 3-D activity. It shows simulated molecules bouncing elastically off each other and the walls of the container. At any point in time, the speed (and energy) of the molecules ranges from zero to high values with the highest probability being towards the low end. (This fairly simple perception works well for consideration of properties such as pressure, temperature and viscosity. A deeper understanding involves probability, molecule configuration, electronic fields and quantum mechanics.

The relaxation time (amount of time that passes between absorption and emission of a photon by a molecule) for CO2 in the atmosphere is about 6 µsec [5, 6]. The elapsed time between collisions between gaseous molecules at sea level average temperature and pressure is about 0.0002 µsec [7]. Thus, at sea level conditions, it is approximately 6/0.0002 = 30,000 times more likely that a CO2 molecule (or any other ghg molecule), after it has absorbed a photon, will bump into another molecule, transferring at least part of the momentum and energy it acquired from the photon. After multiple collisions, essentially all of the added photonic energy becomes distributed among other molecules and the probability of the CO2 molecule emitting a photon at sea level conditions as a direct result of having absorbed one becomes negligible.

The process of sharing the energy with other molecules is thermal conduction in the gas. The process of absorbing photons and sharing the absorbed energy with other molecules is thermalization. Thermalized energy carries no identity of the molecule that absorbed it. Ghg molecules can absorb/emit only photons with certain quanta of energy. Energy itself is quantized at the extremely fine level of 6.626E-34 Js (Planck constant). For all practical purposes, the wavelength spectrum is continuous and there are no forbidden wavelengths of photons. But only terrestrial wavelength photons at about 15 microns can be absorbed/emitted by CO2 molecules.

Emission of electromagnetic radiation from both solid and liquid surfaces of the earth complies with the Planck spectrum (with emissivity ≈0.99) and Stephan-Boltzmann (T4) law. Most particles of clouds, smoke and aerosols emit similarly because they typically contain millions of molecules. Emission of radiation from gas molecules is entirely different (the only gas molecules which can absorb/emit at earth temperatures are ghg molecules i.e. they are IR active). Emission from ghg is quantized and depends on the energy levels of individual molecules. Molecule energy levels are determined probabilistically according to the Maxwell-Boltzmann distribution. The average molecule energy level of the Maxwell-Boltzmann distribution exhibits as the temperature of the gas.

Graphs of the probability distribution curve shape are shown in the Wikipedia article on Maxwell-Boltzmann. A ghg molecule which is jostled to high enough energy for long enough time can emit a photon (of allowed wavelength for that molecule). This is called, for lack of a better term, reverse-thermalization. Reverse-thermalization might alternatively be explained by collisions at adequate energy level between like-species of ghg molecules. The end result at the macro level (emission at low altitude (<3 km) profoundly dominated by water vapor molecules) would look about the same.

Significant terrestrial thermal radiation is nearly all in the wavelength range 6.5-200 microns (1538-50/cm). Water vapor molecules can significantly absorb (and emit) photons over the wavenumber range 0-500/cm compared to CO2. which significantly absorbs over a range of only about 89/cm (625-714/cm). This is displayed in the Hitran2012 [28] generated graphs at Figure 0.4 where minimum intensity is set to 0.0001 to block lines of insignificant intensity.
Figure 0.4: Hitran2012 output for water vapor and carbon dioxide at 296 K, 0.0001 minimum intensity. [28]

Reverse-thermalization, where the warmed jostling molecules (Maxwell-Boltzmann velocity distribution) excite some molecules to emit photons, is almost entirely to water vapor molecules at sea level conditions. There are about 35 times as many water vapor molecules in the atmosphere below about 5 km as there are CO2 molecules (See Figure 2). Each of the water vapor molecules has many high intensity absorb/emit lines where lower energy photons are emitted compared to the low intensity, higher energy, shorter wavelengths for CO2.


Hitran2012 calculator output provides corroborating evidence of the profound dominance of water vapor in atmospheric gas molecule emittance at low altitude. Inputs and results are shown in Figure 0.5.
Figure 0.5: Hitran2012 assessment of carbon dioxide vs water vapor in the atmosphere at ground level.


Figure 0.5 shows many high intensity lines for water vapor at lower energy (lower wavenumber) than the barely discernable indication at wavenumber 667 for carbon dioxide.

Most of the photons emitted by the water vapor molecules are at wavelengths different from the narrow band that CO2 molecules can absorb. Effectively, terrestrial thermal radiation energy absorbed by CO2 near ground level is thermalized and rerouted to high altitude (~10 km ) and/or space via water vapor.

At very high altitudes, temperature, molecule spacing and time between collisions increases to where reverse-thermalization to CO2 molecules becomes significant as does radiation from them to space.


Figure 1 is a typical graph showing top-of-atmosphere (TOA) thermal radiation from the planet. The TOA radiation from different locations on the planet can be decidedly different, e.g. as shown in Figure 9 of Reference [8]. Figure 1, here, might be over a temperate ocean and thus typical for much of earth’s surface. The area under the black trace is about 300 W/m2 which, as expected at low latitude, is somewhat more than the planet average of about 240 W/m2.
Figure 1: Terrestrial thermal radiation and absorption assessed from top-of-atmosphere. Lower wavenumber photons are lower energy.

Typical ‘top-of-atmosphere’ (TOA) emission spectra such as shown in Figure 1 include a ‘notch’ associated with the CO2 absorb/emit band. The existence of this notch demonstrates that CO2 absorbs terrestrial radiation in this wavelength range. Perhaps not as obvious, the presence of the notch also demonstrates thermalization and that the radiation energy which was absorbed by the CO2 molecules was thermalized and substantially redirected to the absorb/emit lines of lower energy (longer wavelength) photons of water vapor molecules. Reverse-thermalization at higher altitudes, where water vapor is greatly reduced, allows some radiation back to CO2. This explains the reduced number of photons at the notch.

An ‘experiment’ demonstrating the effect of reduced water vapor in the atmosphere already exists. Near the poles, the extremely low temperatures result in very low water vapor content while the CO2 level is about the same as everywhere else. With few water vapor molecules available to emit radiation, more of the emission is from CO2 molecules near 15 microns as shown in Figure 9 of Ref 8.

Approximately 98% of dry atmospheric molecules are non-ghg; nearly all nitrogen and oxygen with about 1% argon. They are substantially warmed by thermalization of the photonic energy absorbed by the ghg molecules and, at higher altitudes, cooled by reverse-thermalization back to the ghg molecules.

Figure 2: Water vapor declines rapidly with altitude. [9] (original from NASA)

Thermalized energy carries no identity of the molecule that absorbed it. The thermalized radiation warms the air, reducing its density, causing updrafts which are exploited by soaring birds, sailplanes, and occasionally hail. Updrafts are matched by downdrafts elsewhere, usually spread out but sometimes recognized by pilots and passengers as ‘air pockets’ and micro bursts.

A common observation of thermalization by way of water vapor is cloudless nights cool faster and farther when absolute water vapor content of the atmosphere is lower.

Jostling between gas molecules (observed as temperature and pressure) sometimes causes reverse-thermalization. At low to medium altitudes, EMR emission stimulated by reverse-thermalization is essentially all by way of water vapor.

At altitudes below about 10 km a comparatively steep population gradient (decline with increasing altitude) in water vapor molecules favors outward radiation with increasing amounts escaping directly to space. At higher altitudes, increased molecule spacing and greatly diminished water vapor molecules favors reverse thermalization to CO2. This is observed at and near the sharp peaks at nominal absorb/emit wavelengths of non-condensing ghg (See Figure 1).

Even if the higher scaled intensity (Figure 0.5) of the WV molecules is ignored, a simple calculation shows that doubling the CO2 level has no significant effect on climate. If the minimum intensity is set to 0.0001 to block meaningless ‘noise’, the number of WV lines in the wavenumber range 0-500 is 423 and the number of CO2 lines in the wavenumber range 625-714 is 71 (see Figure 0.4). There are about 35 times as many WV molecules as CO2 molecules so total number of lines is 35*423+71=14876. Doubling the amount of CO2 increases the number of lines by only 71/14876=0.0048 or 0.48%.

Thermalization, the Maxwell-Boltzmann distribution of molecule energy, and quantum mechanics result in ‘climate sensitivity’, the increase in AGT from doubling CO2, to be not significantly different from zero.



Environmental Protection Agency mistakes
The US EPA asserts [10] Global Warming Potential (GWP) is a measure of “effects on the Earth's warming” with “Two key ways in which these [ghg] gases differ from each other are their ability to absorb energy (their "radiative efficiency"), and how long they stay in the atmosphere (also known as their "lifetime").”

The EPA calculation overlooks the very real phenomenon of thermalization. Trace ghg (all ghg except water vapor) have no significant effect on climate because absorbed energy is immediately thermalized.

The EPA calculation of the GWP of a ghg also erroneously overlooks the fact that any added cooling from the increased temperature the ghg might have produced is also integrated over the “lifetime” of the gas in the atmosphere so the duration in the atmosphere ‘cancels out’. Therefore GWP, as calculated by the EPA, egregiously overestimates the influence on average global temperature of noncondensing greenhouse gases. The influence (forcing) of a ghg cannot be more than determined by its immediate concentration. The EPA assessment completely ignores the effect of water vapor which, by far, is the most important ghg and appears to be the only significant ghg.


Water vapor (Rev 8/26/16, 1/11/17, 6/10/17, 9/10/17)
Water vapor is the ghg which makes earth warm enough for life as we know it. Increased atmospheric water vapor contributes to planet warming. Water vapor molecules are far more effective at absorbing terrestrial thermal radiation than CO2 molecules (even if thermalization did not eliminate CO2 as a significant warmer). Humanity’s contribution to atmospheric water vapor increase is primarily (≈ 96%) as a result of increased irrigation (Figure 3.5), with comparatively small contribution from cooling towers at electricity generating facilities. Fossil fuels make an insignificant contribution. Switching to ‘renewables’ will have no significant effect on climate.

Because water vapor is a ghg, increased water vapor causes the planet to warm which further increases vapor pressure of liquid water and therefore increases water vapor so there is a cumulative effect (in control system analysis and electric circuit analysis as done by engineers, this is called positive feedback and is quantified by a dimensionless number which is the ratio of the change with feedback to the change if there was no feedback. The term ‘feedback’ has a different meaning to Climate Scientists and is quantified in units of W/m2). This cumulative effect also increases the rate of cooldown.

Planet warming, as discussed later, increases the vapor pressure of water contributing to the water vapor increase. At present, (thru July 2017) water vapor appears to be increasing about twice as fast as expected based on AGT increase alone. Global temperature increase since 2002 from the UAH trend is about 0.127 K per decade. At 24°C, increase in vapor pressure of liquid water is 5.88% per degree. Percent increase in water vapor due to temperature increase = 0.127 * 5.88% = 0.747%. Measured % increase from TPW in 28 yr = (29.5-28.25)/28.875 = 0.043 = 4.3%. In 10 yr = 10/28*4.3 = 1.54%. Thus measured increase in WV is about 1.54/.747 = 2+ times that for temperature increase alone.

The increased water vapor also causes increased cloud cover which counters temperature increase and will eventually limit it. Sustained increase of only about 1.7% of cloud area would result in an eventual temperature decline of 0.5 °C. Water vapor exhibits a logarithmic decline in absorption effect of equal added increments of water vapor (Fig. 3 of Ref. [12]).

More water vapor in the atmosphere means more warming, acceleration of the hydrologic cycle and increased probability of precipitation related floods. How much of recent flooding (with incidences reported world wide) is simply bad luck in the randomness of weather and how much is because of the ‘thumb on the scale’ of added water vapor?

Essentially all of the ghg effect on earth comes from water vapor. Clear air water vapor measurements over the non-ice-covered oceans in the form of total precipitable water (TPW) have been made since about 1987 by Remote Sensing Systems (RSS) [11]. A graph of this measured ‘global’ average anomaly data, with a reference value of 28.73 added, is shown in the left graph of Figure 3. The slope of the trend is 1.5% increase per decade. The trend of this data is extrapolated both earlier and later using CO2 level as a proxy, with the expression kg/m^2 TPW = 4.5118 * ppmvCO2^0.31286. The result is the right-hand graph of Figure 3 which shows approximately 8% increase since 1960. (The 1940-1950 flat exists in the Law Dome CO2 data base.)
Figure 3: Average clear air total precipitable water over all non-ice-covered oceans. (Rev 8/24/17, 9/19/17)

Clouds (average emissivity about 0.5) consist of solid and/or liquid water particles that radiate approximately according to Planck spectrum and Stephan-Boltzmann (S-B) law (each particle contains millions of molecules).

The perception water vapor content of the atmosphere depends even minutely on CO2 content is profoundly misleading and precisely wrong because it ignores the partial pressure of water vapor caused by (but nearly always less than) the vapor pressure of water.

World Sources of Increased Water Vapor (added 9/10/17, rev 9/25/17)

Irrigation, industrialization, and, increasing population are causing the rise in atmospheric water vapor (WV) above that from feedback (engineering definition of feedback) due to liquid water temperature increase. A survey of available on-line literature provides direct and indirect quantification of significant global sources.

Transportation fuel, linearly interpolated to 2017, amounts to 113E15 BTU/y [31]. Energy content of a typical liquid fuel is 115,000 BTU/gal [32]. Liquid fuels weigh about 6.073 lb/gal = 2.75 kg/gal. Therefore transportation fuels amount to
113E15 * 2.75/115000 = 2.7E12 kg fuel/y                 (a)

About 1.42 kg of WV is produced for each kg of liquid fuel [32] so the amount of WV produced by transportation is
2.7E12 * 1.42 = 3.8E12 kg WV/y                  (b)

World electricity generation is now about 25,000 TWH/y [33]. At an average efficiency of 50% this requires a thermal input of 50,000 TWH/yr. Fuel source fractions of energy [34] interpolated to 2017 are 0.38 coal, 0.36 natural gas and 0.26 non fossil fuel.

Coal combustion produces about 0.4 kg WV/kg coal [35]. Energy content of bituminous coal is about 8200Wh/kg [36]. The amount of WV resulting from burning coal to generate electricity is then
50E15 * 0.38 * 0.4/8200 = 0.93E12 kg WV/y                       (c)

The amount of WV produced by natural gas (nearly all methane, CH4) is readily calculated from the dominant chemical reaction
CH4 + 2O2 => CO2 + 2H2O                (d)

Where a mole of methane weighs about 16 g and the two moles of WV weigh about 18 g each.
Natural gas energy content is about 15,400 Wh/kg [36]. The amount of WV resulting from burning natural gas to generate electricity is then
50E15 * 0.36 *36/16/15400 = 2.6E12 kg WV/y                    (e)

The total WV from all fossil fuel used to generate electricity is then
0.9E12 + 2.6E12 = 3.5E12 kg WV/y                         (f)

Waste energy during electricity generation can be approximately accounted for by evaporation of water in cooling towers, etc. At 50% efficiency the waste energy is the same as the energy in the electricity produced, 25,000 TWH/yr = 25E12 kWh/y.
Latent heat of water = 2257 kJ/kg = 0.627 kWh/kg = 1.594 kg/kWh.
The amount of WV from waste heat (cooling tower, etc.) during electricity generation is then
25E12 * 1.594 = 39.8E12 kg WV/y                           (g)

Irrigation is by far the largest source of WV. The increase in irrigation is indicated by the increase in withdrawal for agriculture as shown in Figure 3.5 [37].
Figure 3.5: Global water withdrawal includes both ground water and surface water [37]

The total agricultural area equipped for irrigation in 2009 was 311E10 m2 of which 84% were actually being irrigated [38]. Estimating an increase of 2% to 2017, the total area being irrigated is now about
311E10 * 0.84 * 1.02 = 266E10 m2                            (h)

Total annual fresh water withdrawal (both ground and surface) is now 3,986 km3 = 3.986E15 kg/y [39]. Of this, about 70% is for agricultural use [40]. This works out to
3.986E15 * 0.7/266E10 = 1052 kg/m2/y ≈ 1 m/y                     (i)
which appears reasonable because average rainfall for the planet is about 1 m/y.

Evapotranspiration, WV from plants and landscape, is discussed in the ‘thematic discussion’ of Aquastat [37]. From there, the amount of precipitation on land is 110,000 km3 of which the fraction evapotranspirated is 0.56 + 0.05 = 0.61. Given the planet surface area of 510.1E6 km2, and land fraction of 0.29 this results in the equivalent liquid depth of the total amount of water leaving the surface as WV as
110,000 * 0.61/0.28/510.1E6 = 0.00047 km = 0.47 m                       (j)

Water weighs 1000 kg/m3 so evapotranspiration amounts to 470 kg/m2

Approximately 95% of the irrigated area is flood irrigated so, to simplify calculation, assume all irrigation is flood irrigation approximated as furrow type [41]. Optimum frequency is to flood the furrows about every 10 days [42]. Thus about half the area is covered by water 10% of the time where evaporation from the water is about one meter per year [43] and the rest of the time, the additional evaporation is assumed to be according to the calculated evapotranspiration. Evapotranspiration prior to irrigation must have been low or irrigation would not be done. Evapotranspiration with irrigation, to be cost effective, is most likely to be much more than calculated. These two uncertainties are assumed to approximately cancel each other. The total amount of WV resulting from irrigation is then
(0.1 * (1 + 0.47)/2 + 0.9 * 0.47) * 266E10 = 132.1E10 m3 = 132.1E13 kg/y                        (k)

These calculations are summarized in Table 0
Water vapor source
E13 kg/y
Irrigation
132.1
Transportation fuel
0.4
Fossil fuel for electricity generation
0.4
Cooling towers, etc. for electricity generation
4.0
Total
136.9
Table 0: Summary of contributions to atmospheric water vapor.

Approximately 132.1/136.9 = 0.96+ or 96+% of atmospheric WV increase above that due to feedback (engineering definition of feedback) from liquid water temperature increase results from irrigation. 


The AGT Model
Most modeling of global climate has been with Global Climate Models (GCMs) where physical laws are applied to a 3-dimensional grid consisting of hundreds of thousands of discrete blocks and the interactions between the discrete blocks are analyzed using super computers with an end result being calculation of the AGT trajectory. This might be described as a ‘bottom up’ approach. Although theoretically promising, multiple issues currently exist with this approach. Reference [13] discloses that nearly all of the more than 100 current GCMs are obviously faulty. The few which appear to follow measurements might even be statistical outliers of the ‘consensus’ method. The growing separation between calculated and measured AGT as shown at Figure 9 in Ref. [14] also suggests some factor is missing.

The approach in the analysis presented here is ‘top down’. This type of approach has been called ‘emergent structures analysis’. As described by Dr. Roy Spencer in his book The great global warming Blunder, “Rather than model the system from the bottom up with many building blocks, one looks at how the system as a whole behaves.” That approach is used here with strict compliance with physical laws.

The basis for assessment of AGT is the first law of thermodynamics, conservation of energy, applied to the entire planet as a single entity. Much of the available data are forcings or proxies for forcings which must be integrated (mathematically as in calculus, i.e. accumulated over time) to compute energy change. Energy change divided by effective thermal capacitance is temperature change. Temperature change is expressed as anomalies which are the differences between annual averages of measured temperatures and some baseline reference temperature; usually the average over a previous multiple-year time period. (Monthly anomalies, which are not used here, are referenced to previous average for the same month to account for seasonal norms.)

The AGT model, a summation of contributing factors, is expressed in this equation:

Tanom = (A,y)+thcap-1 * Σyi=1895 {B*[S(i)-Savg] + C*ln[TPW(i)/TPW(1895)] –                              F * [(T(i)/T(1895))4 – 1]} + D                                                                                 (1)

Where:
Tanom = Calculated average global temperature anomaly with respect to the baseline of the anomaly for the measured temperature data set, K
A = highest-to-lowest extent in the saw-tooth approximation of the net effect on planet AGT of all ocean cycles, K
y = year being calculated
(A,y) = value of the net effect of ocean cycles on AGT in year y (α-trend), K
thcap = effective  thermal capacitance [1] of the planet = 17±7 W yr m-2 K-1
1895 = Selected beginning year of acceptably accurate world wide temperature measurements.
B = combined proxy factor and influence coefficient for energy change due to sunspot number anomaly change, W yr m-2
S(i) = average daily V2 sunspot numbers [15,16] in year i
Savg = baseline for determining SSN anomalies
C = influence coefficient for energy change due to TPW change, W yr m-2
TPW(i) = total precipitable water in year i, kg m-2  (from calculation for Fig 3)
TPW(1895) = TPW in 1895, same units as TPW(i)  (from calculation for Fig 3)
F = 1 to account for change to S-B radiation from earth due to AGT change, W yr m-2
T(i) = AGT calculated by adding T(1895) to the reported anomaly, K
T(1895) = AGT in 1895 = 286.707 K
D = offset that shifts the calculated trajectory vertically on the graph, without changing its shape, to best match the measured data, K (equivalent to changing the anomaly reference temperature).

Accuracy of the model is determined using the Coefficient of Determination, R 2, to compare calculated AGT with measured AGT.


Approximate effect on the planet of the net of ocean surface temperature (SST)
The average global ocean surface temperature oscillation is only about ±1/6 K. It is defined to not significantly add or remove planet energy. The net influence of SST oscillation on reported AGT is defined as α-trend. In the decades immediately prior to 1941 the amplitude range of the trends was not significantly influenced by change to any candidate internal forcing effect; so the observed amplitude of the effect on AGT of the net ocean surface temperature trend anomaly then, must be approximately the same as the amplitude of the part of the AGT trend anomaly due to ocean oscillations since then. This part is approximately 0.36 K total highest-to-lowest extent with a period of approximately 64 years (verified by high R2 in Table 1).

The measured AGT trajectory (Figure 9) suggests that the least-biased simple wave form of the effective ocean surface temperature oscillation is approximately saw-toothed. Approximation of the sea surface temperature anomaly oscillation can be described as varying linearly from –A/2 K in 1909 to approximately +A/2 K in 1941 and linearly back to the 1909 value in 1973. This cycle repeats before and after with a period of 64 years.

Because the actual magnitude of the effect of ocean oscillation in any year is needed, the expression to account for the contribution of the ocean oscillation in each year to AGT is given by the following:

ΔTosc = (A,y)             K (degrees)                 (2)

where the contribution of the net of ocean oscillations to AGT change is the magnitude of the effect on AGT of the surface temperature anomaly trend of the oscillation in year y, and A is the maximum highest-to-lowest extent of the effect on AGT of the net ocean surface temperature oscillation.

Equation (2) is graphed in Figure 4 for A=0.36.

Figure 4: Ocean surface temperature oscillations (α-trend) do not significantly affect the bulk energy of the planet.


Comparison of approximation with ‘named’ ocean cycles
Named ocean cycles include, in the Pacific north of 20N, Pacific Decadal Oscillation (PDO); in the equatorial Pacific, El Nino Southern Oscillation (ENSO); and in the north Atlantic, Atlantic Multidecadal Oscillation (AMO).

Ocean cycles are perceived to contribute to AGT in two ways: The first is the direct measurement of sea surface temperature (SST). The second is warmer SST increases atmospheric water vapor which acts as a forcing and therefore has a time-integral effect on temperature. The approximation, (A,y), accounts for both ways.

SST data is available for three named cycles: PDO index, ENSO 3.4 index and AMO index. Successful accounting for oscillations is achieved for PDO and ENSO when considering these as forcings (with appropriate proxy factors) instead of direct measurements. As forcings, their influence accumulates with time. The proxy factors must be determined separately for each forcing. The measurements are available since 1900 for PDO [17] and ENSO3.4 [18]. This PDO data set has the PDO temperature measurements reduced by the average SST measurements for the planet.

The contribution of PDO and ENSO3.4 to AGT is calculated by:
PDO_NINO = Σyi=1900 (0.017*PDO(i) + 0.009 * ENSO34(i))        (3)

Where:
            PDO(i) = PDO index [17] in year i
            ENSO34(i) = ENSO 3.4 index [18] in year i

How this calculation compares to the idealized approximation used in Equation (2) with A = 0.36 is shown in Figure 5.


Figure 5: Comparison of idealized approximation of ocean cycle effect and the calculated effect from PDO and ENSO.

The AMO index [19] is formed from area-weighted and de-trended SST data. It is shown with two different amounts of smoothing in Figure 6 along with the saw-tooth approximation for the entire planet per Equation (2) with A = 0.36.
Figure 6: Comparison of idealized approximation of ocean cycle effect and the AMO index.

The high Coefficients of Determination in Table 1 and the comparisons in Figures 5 and 6 corroborate the assumption that the saw-tooth profile with a period of 64 years provides adequate approximation of the net effect of all named and unnamed ocean cycles in the calculated AGT anomalies.

Atmospheric carbon dioxide (rev 1/11/17)
The level of atmospheric carbon dioxide (CO2) has been widely measured over the years. Values from ancient times were determined by measurements on gas bubbles which had been trapped in ice cores extracted from Antarctic glaciers [20]. Spatial variations between sources have been found to be inconsequential [2]. The best current source for atmospheric carbon dioxide level [21] is Mauna Loa, Hawaii. The left graph in Figure 7 provides insight as to the fraction of atmospheric CO2 for various times and conditions. The planet came perilously close to extinction of all plants and animals due to the low level of CO2 at the end of the last glaciation. For plant growth, even at the current level the atmosphere is impoverished for CO2.

Figure 7: Atmospheric carbon dioxide levels.


Extrapolation to future CO2 levels, shown in the right side graph in Figure 7, is accomplished using a second-order curve fit to data measured at Mauna Loa from 1980 to 2012. Although CO2 has no significant effect on climate, the trajectory shape, including data back to 1610 from Law Dome (275 ppmv), was used as a proxy to extrapolate TPW back to 1610. 

Sunspot numbers
Sunspots have been regularly recorded since 1610. In 2015 historical (V1) SSN were reevaluated in light of current perceptions and more sensitive instruments and are designated as V2. The V2 SSN data set is used throughout this assessment. V2 SSN [15] are shown in Figure 8.

Sunspot numbers (SSN) are seen to be in cycles each lasting approximately 11 years. The current cycle, called 24, has been comparatively low, has peaked, and is now in decline.

The Maunder Minimum (1645-1700), an era of extremely low SSN, was associated with the Little Ice Age. The Dalton Minimum (1790-1820) was a period of low SSN and low temperatures. An unnamed period of low SSN (1880-1930) was also accompanied by comparatively low temperatures.

An assessment of this is that sunspots are somehow related to the net energy retained by the planet, as indicated by changes to the average global temperature trend. Fewer sunspots are associated with cooling, and more sunspots are associated with warming. Thus the hypothesis is made that SSN are proxies for the rate at which the planet accumulates (or loses) radiant energy over time. Therefore the time-integral of the SSN anomalies is a proxy for most of the amount of energy retained by the planet above or below breakeven.

Also, a lower solar cycle over a longer period might result in the same increase in energy retained by the planet as a higher solar cycle over a shorter period. Both magnitude and time are accounted for by taking the time-integral of the SSN anomalies, which is simply the sum of annual mean SSN (each minus Savg) over the period of study.

SSN change correlates with change to Total Solar Irradiance (TSI). However, TSI change can only account for less than 10% of the AGT change on earth. Because AGT change has been found to correlate with SSN change, the SSN change must act as a catalyst on some other factor (perhaps clouds [22]) which have a substantial effect on AGT.


Figure 8: V2 SSN [15]


Possible values for Savg are subject to two constraints. Initially they are determined as that which results in derived coefficients and maximum R2. However, calculated values must also result in rational values for calculated AGT at the depths of the Little Ice Age. The necessity to calculate a rational LIA AGT is a somewhat more sensitive constraint. The selected value for Savg results in calculated LIA AGT of approximately 1 K less than the recent trend which appears rational and is consistent with most LIA AGT assessments.


AGT measurement data set
In the last few years, reported temperature data, especially land temperature data, have been changed by the reporting agencies. This detracts from their applicability in any correlation.

Rapid year-to-year changes in reported temperature anomalies are not physically possible for true energy change of the planet. The sharp peak in 2015, which coincides with an extreme El Nino, is especially distorting. It, at least in part, will be compensated for by a La Nina which is likely to follow. For analysis here, the El Nino spike is compensated for by replacing reported AGT for 2013-2015 with the average 2002-2012.

 A further bit of confusion is introduced by satellite determinations. Anomalies they report as AGT anomalies are actually for the lower troposphere (LT), have a different reference temperature (reported anomalies determined using satellite data are about 0.2 K lower), and appear to be somewhat more volatile (about 0.15 K further extremes than surface measurements) to changes in forcing.

The data set used for this assessment is the current (5/27/16) HadCRUT4 data set [23] through 2012 with 2013-2015 set at the average 2002-2012 at 0.4863 K above the reference temperature. This set is shown in Figure 9.

Figure 9: HadCRUT4 data set as of 5/27/16 with flat starting in 2013 as used here.


The sunspot number anomaly time-integral is a proxy for a primary driver of the temperature anomaly β-trend
By definition, energy change divided by effective thermal capacitance is temperature change.

In all cases in this document, coefficients (A, B, C, & D) which achieved maximum R2 for unsmoothed data sets were not changed when calculating R2 for smoothed data. F=1 for all cases.

Incremental convergence to maximum R2 is accomplished by sequentially and repeatedly adjusting the coefficients. The process is analogous to tediously feeling the way along a very long and narrow mathematical tunnel in 4-dimensional mathematical space. The ‘mathematical tunnel’ is long and narrow because the influence on AGT determined by the SSN anomaly time-integral, at least until the last decade or so, is quite similar to the influence on AGT as determined by the rise in TPW.

Measured temperature anomalies in Figure 10 are HadCRUT4 data as shown in Figure 9. The excellent match of the up and down trends since before 1900 of calculated and measured temperature anomalies, shown here in Figure 10, and, for 5-year moving average smoothed temperature anomaly measurements, in Figure 11, demonstrate the usefulness and validity of the calculations. All reported values since before 1900 are within the range ±2.5 sigma (±0.225 K) from the calculated trend. Note: The variation is not in the method, or the measuring instruments themselves, but results from the effectively roiling (at this tiny magnitude of temperature change) of the object of the measurements. [44]

Projection until 2020 uses the expected sunspot number trend for the remainder of solar cycle 24 as provided [16] by NASA. After 2020 the ‘limiting cases’ are either assuming sunspots like from 1924 to 1940 or for the case of essentially no sunspots which is similar to the Maunder Minimum.

Some noteworthy volcanoes and the year they occurred are also shown on Figure 10. No consistent AGT response is observed to be associated with these. Any global temperature perturbation that might have been caused by volcanoes of this size is lost in the natural fluctuation of measured temperatures.

Much larger volcanoes can cause significant temporary global cooling from the added reflectivity of aerosols and airborne particulates. The Tambora eruption, which started on April 10, 1815 and continued to erupt for at least 6 months, was approximately ten times the magnitude of the next largest in recorded history and led to 1816 which has been referred to as ‘the year without a summer’. The cooling effect of that volcano exacerbated the already cool temperatures associated with the Dalton Minimum.

Figure 10: Measured average global temperature anomalies with calculated future trends using Savg = 60 and with V2 SSN. R 2 = 0.904520. (Rev 8/26/16)

Coefficients in Equation (1) which were determined by maximizing R2 identify maximums for each of the factors explicitly considered. Factors not explicitly considered (such as unaccounted for residual (apparently random) variation in reported annual measured temperature anomalies, aerosols, CO2, other non-condensing ghg, volcanoes, ice change, etc.) must find room in the unexplained residual, and/or by occupying a fraction of the effect otherwise occupied by each of the factors explicitly considered. 

Figure 11: Same as Figure 10 but with 5-year running average of measured temperatures. R2 = 0.981782. (Rev 8/26/16)

The derived coefficients and other results are summarized in Table 1. Note that a coefficient of determination, R2 = 0.981782 means a near-perfect correlation coefficient of 0.99.

The influence of the net effect of factors other than the net effect of ocean cycles on AGT can be calculated by excluding the α-trend (set 'A' to zero) from the AGT which was calculated using Equation (1). For the values used in Figure 10, this results in the β-trend as shown in Figure 12. Note that in 2005 the anomaly from other than α-trend, as shown in Figure 12, is A/2 lower than the calculated trend in Figures 10 and 11 as it should be.


Figure 12: Anomaly trend (β-trend). Equation (1) except summation starts at i = 1610 and excluding α-trend. (Rev 8/26/16)


How the β-trend could take place
Although the connection between AGT and the sunspot number anomaly time-integral is demonstrated, the mechanism by which this takes place remains somewhat speculative.

Various papers have been written that indicate how the solar magnetic field associated with sunspots can influence climate on earth. These papers posit that decreased sunspots are associated with decreased solar magnetic field which decreases the deflection of and therefore increases the flow of galactic cosmic rays on earth.

Henrik Svensmark, a Danish physicist, found that increased flow of galactic cosmic rays on earth caused increased low altitude (<3 km) clouds and planet cooling. An abstract of his 2000 paper is at [24]. Marsden and Lingenfelter also report this in the summary of their 2003 paper [25] where they make the statement “…solar activity increases…providing more shielding…less low-level cloud cover… increase surface air temperature.” These findings have been further corroborated by the cloud nucleation experiments [26] at CERN.

These papers [24, 25] associated the increased low-altitude clouds with increased albedo leading to lower temperatures. Increased low altitude clouds would also result in lower average cloud altitude and therefore higher average cloud temperature. Although clouds are commonly acknowledged to increase albedo, they also radiate energy to space so increasing their temperature increases S-B radiation to space which would cause the planet to cool. Increased albedo reduces the energy received by the planet and increased radiation to space reduces the energy of the planet. Thus the two effects work together to change the AGT of the planet.

A contributing or possibly alternate speculation is that clouds might also be affected by solar wind. End result is the same: Average global temperature correlates with the time-integral of sunspot number anomalies (when combined with two other factors as shown in Equation (1)).

Simple analyses [22] indicate that either an increase of approximately 186 meters in average cloud altitude or a decrease of average albedo from 0.3 to the very slightly reduced value of 0.2928 would account for all of the 20th century increase in AGT of 0.74 K. Because the cloud effects work together and part of the temperature change is due to ocean oscillation (low in 1901, 0.2114 higher in 2000), substantially less cloud change would suffice.


Hind Cast Estimate
Average global temperatures were not directly measured in 1610 (accurate thermometers had not been invented yet). Recent estimates, using proxies, are few. The temperature anomaly trend that Equation (1) calculates for that time is roughly consistent with other estimates. The decline in the trace 1615-1715 on Figure 12 results from the low sunspot numbers for that period as shown on Figure 8.

As a possibility, the period and amplitude of oscillations attributed to ocean cycles demonstrated to be valid after 1895 are assumed to maintain back to 1610. Equation (1) is modified to begin integration in 1610. The coefficient D is changed to make the calculated temperature in 2005 equal to what it is in Figure 10.

Temperature anomalies thus calculated, estimate possible trends since 1610 and actual trends of reported temperatures since they have been accurately measured world wide.  This assessment is shown in Figure 13.


Figure 13: Calculated temperature anomalies using Equation (1) with the same coefficients as for Figure 10 and V2 SSN. Measured temperature anomalies from Figure 9, and anomaly range estimates determined by Loehle are superimposed. (Rev 8/26/16)


A survey [27] of non-tree-ring global temperature estimates was conducted by Loehle including some for a period after 1610. Simplifications of the 95% limits found by Loehle are also shown on Figure 13. The spread between the upper and lower 95% limits are fixed, but, since the anomaly reference temperatures might be different, the limits are adjusted vertically to approximately bracket the values calculated using Equation (1). The fit appears reasonable considering the uncertainty of all values.

Calculated temperature anomalies look reasonable back to 1700 but indicate higher temperatures prior to that than most proxy estimates. They are, however, consistent with the low sunspot numbers in that period. They qualitatively agree with Vostok, Antarctica ice core data but decidedly differ from Sargasso Sea estimates during that time (see the graph for the last 1000 years in Reference [2]). Worldwide assessments of average global temperature, that far back, are sparse and speculative. Ocean oscillations might also have been different from assumed.


Projection from 1990
Figure 14 shows the calculation using Equation (1) with coefficients determined using HadCRUT4 measured temperatures to 1990. The calculated AGT trend in 2020 projected from  1990 is 0.06 K cooler than the projection from 2015.

Figure 14: Same as Figure 11 except coefficients determined using data through 1990.


Step changes in AGT
Interpretation of a reported sudden AGT increase (or decrease) as planet energy increase (or decrease) is physically impossible because of the huge effective thermal capacitance which results in a 5-year time constant [1] for thermal response of the planet to a step change in forcing.


Influence of atmospheric water vapor on AGT
The temperature increase through 2015 attributable to TPW is the net of the increase from TPW and the decrease from added S-B radiation due to the part of the temperature rise attributable to TPW which is above the 1895 value of 286.707 K. The net effect is designated ΔTTPW.

At least until the last decade or so, the influence on AGT due to TPW has been quite similar to the influence on AGT determined by the SSN anomaly time-integral. This similarity has resulted in the effect of TPW being erroneously masked by the calculated effect of sunspot number anomalies.

Figure 15 shows how increasing water vapor has contributed to AGT. It is the same as Figure 11 but shows also the calculated trajectory if there had been no increase in water vapor since 1895. This is calculated by setting C to zero and retaining the other coefficients in Equation (1).

It is speculated that local conditions might result in a local thermal runaway which is observed as a super el Niño. The sharp spike and following la Nina are consistent with this hypothesis.

 
Figure 15: Same as Figure 11 but with calculated trajectory incorporated for the case if there was no increase in water vapor since 1895. (added 10/31/16)

Values for the coefficients and results are summarized in Table 1.


Table 1: A, B, C, D, F refer to coefficients in Equation 1. The column headed # is a code identifying the particular EXCEL file used. (Rev 8/26/16)
Conclusions
Three factors explain essentially all of AGT change since before 1900. They are ocean cycles, accounted for with an approximation, influence quantified by a proxy which is the SSN anomaly time-integral and, the gain in atmospheric water vapor measured since 1987 and extrapolated before and after using measured CO2 as a proxy.

Others have looked at only amplitude or only duration factors for solar cycles and got poor correlations with average global temperature. As typically done, this procedure violates the relation between math and the physical world. The excellent (and computationally valid) correlation comes by combining the two, which is what the time-integral of sunspot number anomalies does. Prediction of future sunspot numbers more than a decade or so into the future has not yet been confidently done. 

As displayed in Figure 12, the β-trend shows the estimated true average global temperature trend (the net average global energy trend) during the planet warm up from the depths of the Little Ice Age.

The net effect of ocean oscillations is to cause the surface temperature α-trend to oscillate above and below the β-trend. Equation (1) accounts for both trends.

Figure 11 shows the near perfect match with calculated temperatures which occurs when random fluctuation in reported measured temperatures is smoothed out with 5-year moving average.

Warming attributed to increasing water vapor explains the flat measured AGT trend in spite of declining sunspot and ocean cycle forcings and might delay or even prevent global cooling.

The increasing trend of global average water vapor as shown in Figure 3, besides countering the temperature decline which would otherwise be occurring, is a likely contributor to increased precipitation and flooding.

The measured water vapor increase is approximately twice expected from liquid water temperature increase alone.

Long term prediction of average global temperatures depends substantially on long term prediction of sunspot numbers.


References: (rev 10/21/16, 9/10/17, 9/20/17, 9/25/17, 10/20/17)
1. Effective thermal capacitance & time constant: Schwartz, Stephen E., (2007) Heat capacity, time constant, and sensitivity of earth’s climate system, J. Geophys. Res., vol. 113, Issue D15102, doi:10.1029/2007JD009373 
2. 2008 assessment of non-condensing ghg  http://www.middlebury.net/op-ed/pangburn.html
5. 6 microsecond relaxation time in atmosphere  http://onlinelibrary.wiley.com/doi/10.1002/qj.49709540302/abstract                                       10 microsecond CO2 relaxation in atmosphere: https://www.reddit.com/r/climateskeptics/comments/14hvl9/ucar_presents_a_cartoon_to_misrepresent_what/    https://lofi.physforum.com/Greenhouse-Gas-Effect-and-Carbon-Dioxide_7157.html
6. 7.1 microsecond CO2 relaxation in pure gas http://pubs.rsc.org/en/Content/ArticleLanding/1967/TF/TF9676302093#!divAbstract ).      
11. NASA/RSS TPW (they only report the latest month available, 08 means August) http://data.remss.com/vapor/monthly_1deg/tpw_v07r01_198801_201708.time_series.txt
14. Analysis with V1 SSN sans water vapor: Pangburn 2014, Energy & Environment V25, No. 8 1455-1471
15. V2 sunspot numbers http://www.sidc.be/silso/datafiles
16. Graphic of V2 Solar cycle 24: http://solarscience.msfc.nasa.gov/predict.shtml
20. CO2 level at Law Dome, Antarctica: http://cdiac.ornl.gov/ftp/trends/co2/lawdome.combined.dat
22. Sensitivity of AGT to clouds http://lowaltitudeclouds.blogspot.com
24. Svensmark paper: Phys. Rev. Lett. 85, 5004–5007 (2000) http://prl.aps.org/abstract/PRL/v85/i23/p5004_1
25. Marsden & Lingenfelter 2003, Journal of the Atmospheric Sciences 60: 626-636 http://www.co2science.org/articles/V6/N16/C1.php
29. Relevant issues and consensus mistakes http://consensusmistakes.blogspot.com
34. Fuel sources for electricity generation https://www.eia.gov/outlooks/ieo/electricity.php
36. Energy content of bituminous coal https://en.wikipedia.org/wiki/Energy_density
39. Annual fresh water withdrawal https://data.worldbank.org/indicator/ER.H2O.FWTL.K3
40. 70% of withdrawal is for agriculture https://data.worldbank.org/indicator/er.h2o.fwag.zs
42. Frequency of furrow irrigation https://naldc.nal.usda.gov/download/54786/PDF

44. Animation of roiling SST https://www.youtube.com/watch?v=aKMY4JRN0kk