Negative Feedback Prevents Harmful Temperatures from Carbon Dioxide
Guest essay by Bryce Johnson
The purpose of the article is to contribute to refuting the false alarm that has been generated worldwide about excess global warming caused by atmospheric carbon dioxide, primarily by mischaracterizing its inherently negative feedback.
Feedback is the effect that the result of a process has on itself. Positive feedback augments the result and negative feedback diminishes it. A negative feedback does not preclude an increase in the results of a process but it does limit the ultimate magnitude of the increase. Most feedbacks in nature are negative or soon become negative. Otherwise we would have been destroyed by nature some time ago. Positive feedbacks can cause runaway effects. Nature abhors a positive feedback.
A mathematical definition of feedback is derived from the following schematic.
Classical Electrical Circuit Feedback
In the schematic, it is noted that the feedback can change either the input or the process to alter the output. Without the feedback loop connected the output depends only on the process. But when connected that makes the new output 1+ f which is, in turn, operated on by the “f” factor which when added to the input signal, 1, produces an output of 1 + f + f2 and a subsequent one of 1 +f(1 + f + f2). Since the process is continuous the ultimate result is
1 + f + f2 + . . . . . . +fn
as n approaches infinity. Its sum is expressed as a single term, 1/ (1-f) if f is between 0 and +1. If f is negative the output oscillates but the oscillations decrease and a lower final constant output is eventually achieved
So final output signal, S, is
S = A/ (1-f) (1)
where A is the output with no feedback. This is the standard expression for electrical circuit feedback. If f approaches +1, a “tipping point” is reached where the amplification attempts to become infinite. Such a tipping point is the basis of the warming advocates’ claim of a runaway catastrophic warming which is frequently cited as evidence for their climate-change alarm (1). But negative feedback prevents such a runaway.
The global warming analog to the feedback schematic entails CO2 insertion in the atmosphere as “input,” its absorption of infrared radiation (IR) as the “process” and altered atmospheric temperature as the “output.” Two commonly recognized feedbacks from the increase of CO2 in the atmosphere are:
1) Saturation whereby CO2 absorption diminishes the energy content of the emitted IR leaving subsequent CO2 additions with less energy to absorb which diminishes the overall absorption per molecule. Saturation affects the process and all greenhouse gases saturate.
2) Water vapor from surface evaporation caused by the CO2 warming that can change to liquid or ice phases and can theoretically increase warming because water, like CO2, exhibits a greenhouse effect in all of its phases. But both its production and its presence also cool the atmosphere. Water in the atmosphere in any of its three phases is a universal inhibitor of global warming, yet it is totally mischaracterized by the global warming community as having a significant positive feedback.
Despite evidence to the contrary there is a considerable body of opinion that water-vapor feedback is positive. But the negativity of the saturation feedback is not questioned.
A third, but rarely acknowledged, feedback is:
3) Removal of CO2 from the atmosphere. It is shown in this article that the atmospheric CO2 level fails to keep up with the amount inserted therein. The fraction which is permanently removed quickly surpasses the fraction which remains and its increase continues after insertion stops until equilibrium is achieved among the carbon reservoirs (atmosphere, land and ocean). There are a number of reports that compare measured time-dependent input CO2 with corresponding increased CO2 level in the atmosphere. Most of these report that the atmosphere loses about fifty percent of what is inserted (less than that calculated here). However, quoting directly from Reference 2: “Over the long term (millennial timescales), the ocean has the potential to take up approximately 85 percent of the anthropogenic CO2 that is released to the atmosphere.” And this excludes any possible permanent removal by soil and plants. This analysis shows that soil and plants remove much of it in the short term. By the analysis of this article, removal has a significant negative impact (illustrated in Figure 10). The world carbon balances typified by the three in Appendix A are all close to that of the Woods Hole Balance of Figure A-3 which shows 100 petagrams (pg) of carbon exchanged each year with both the ocean and the land with its vegetation.
There is yet another feedback source and is obviously positive but its magnitude is negligible compared to the above three and that is production of CO2 by its out-gassing from a ghg-warmed ocean. The insignificance of its magnitude is demonstrated under CALCULATIONS: Out-gassing of Ocean CO2
The negativity of the main three feedbacks is shown graphically in Figures 1, 2 and 3. Figures 1 and 2 are illustrative, only, to portray negative feedback. But Figure 3 results are actually used in the calculations of predicted maximum temperature rise. The rate of change of negative feedback typically diminishes with time but the overall decrease in output never ceases, as shown in the figures. This is the well-known asymptotic change which continually approaches, but never actually reaches a constant limiting value. These three mechanisms are the major feedback mechanisms; they originate completely within the atmosphere and they directly involve CO2, the object of the warming concern.
Forcing functions of Figures 1 and 2, in watts per square meter deposited in the atmosphere (the driver of warming) per added unit CO2 or water vapor are calculated by the Modtran computer code described in Appendix B. These are proportional to temperature rise produced. Figures 1 and 2 demonstrate the universality of the negative feedbacks of both CO2 and water vapor. For any combination of atmospheric content of CO2 and water vapor in each figure THE INCREMENTAL TEMPERATURE FORCING FUNCTION IS REDUCED BY INCREASING THE CONTENT OF EITHER. This is an important observation because it defines negative feedback. Reduction of the incremental forcing function does not eliminate the forcing function, as shown in Figures 5 and 6, but it does place a finite upper bound on it. A recent study at the University of Alabama at Huntsville (UAH) confirms this negativity of feedback from atmospheric carbon (3).
The negative impact of atmosphere removal is best illustrated by plotting the fraction remaining of atmospheric CO2 input versus time as in Figure 3.
The CO2 input to the atmosphere ceases at 1000 years when fossil fuels with their carbon content is determined to be depleted by the model of this article. CO2 removal continues until equilibrium of all the biosphere reservoirs is reached. Figure 3 indicates a variable rate in the increase of the removal and the removal continues until the biosphere components equilibrate. Figure 10 indicates the relative contribution of the biosphere’s carbon reservoirs to this removal.
Negative feedbacks normally reach a constant limit asymptotically. CO2 removal achieves it when the biosphere reservoirs reach equilibrium following cessation of CO2 input, which is assured by the assumption of this model that reservoir expulsion rate is proportional to its content. That feedback, however, can actually continue at a very low level as shown in Figures A1 and A2 wherein some carbon from the atmosphere is shown to be continually and permanently deposited on the ocean floor.
Water vapor feedback exactly fits the feedback alteration of the input shown in the feedback schematic. The greenhouse-gas (ghg) atmospheric heat from CO2 produced water vapor can also heat the atmosphere by its own ghg effect. But its production removes more heat from the atmosphere than its added ghg effect can replace. Its presence also cools the atmosphere (see Figures 5, 6 and 7). It has its own negative saturation feedback and also augments CO2’s negative saturation feedback. The analysis presented here shows that it has a universal and significant negative feedback.
Saturation doesn’t change the ghg input but it does alter the “process” negatively by reducing the heating effect of the individual molecules.
Removal obviously reduces the number of input molecules as shown in Figure 3 and it reduces maximum possible temperature increase from 2.45 down to 0.98 oC.
Position of the International Panel on Climate Chance (IPCC) and Associated Catastrophic Athropogenic Global Warming (CAGW) Adherents
The IPCC has generated world-wide clamor to curtail carbon dioxide (CO2) production in order to prevent alleged CAGW which relies on the claim that significant positive temperature feedback exists from adding CO2, in contrast to the evidence presented above. Without such a claim there would be no case for a runaway temperature, or “tipping point” (1). Negative feedback does not prevent CO2-caused temperature rise but typical IPCC results, as shown in Figure 4, significantly over-predict measured temperature increase. This article presents defensible data with a ‘limit analysis’ (described below under CALCULATIONS) showing that CO2 feedback is sufficiently negative to preclude unacceptable warming. IPCC calculations and explanations of their results are not consistent with physical science.
Figure 4 shows a comparison of predictions and measurements of world temperature increase for this century. The blue curve is from an average provided by Roy Spencer and John Christy at the University of Alabama at Huntsville (UAH) of 44 separate calculations of warming by various researchers (4) and noted by Spencer that “it approximately represents what the IPCC uses for its official best estimate of projected warming.” The pink curve is by converting the net forcing function across the atmosphere from added CO2 as calculated by the Modtran computer code (described in Appendix B) to temperature increase. The conversion formula is described under CALCULATIONS: Temperature Rise. The input CO2 values used for this method are from the well-known measurements at Mauna Loa, commonly called the “Keeling” curve (5) whose accuracy is universally accepted. UAH regularly publishes the national satellite temperature measurements made. The yellow curve is from UAH satellite data and the green curve is from surface measurements by The Goddard Institute for Space Studies (GISS). These two curves are the results of a linearized least-squares fit to the actual data (6).
The average of the 44 values from the IPCC contributions exceeds the average of the measured values by a factor of seven, but the computation method of this article matches them very closely.
Summary of IPCC/CAGW Problems
1. The seeds of the IPCC’s global warming bias were planted in its UN charter: i.e., to assess the scientific basis of human-induced climate change (7). Any finding of an insignificant scientific basis would end the IPCC and its promotion of munificent research contracts.
2. As indicated above, the negative feedback for CO2 insertion precludes the possibility of the claimed “tipping point.” IPCC’s temperature predictions are much too high (Figure 4). IPCC is a primary source of unwarranted CAGW alarm.
3. The well publicized emails from contributors to IPCC research known as the “climategate” scandal of November 2009 (8) revealed undeniable scientific malfeasance in preventing publication of dissent from their theories, in concealing their methods and in data manipulation as well. Unfortunately, these are still major strategies employed in promoting CAGW alarm, but mere stifling of dissent messages has now been elevated to attempted destruction of the careers of dissenters.
4. IPCC’s 5th assessment report (AR5) shows lower temperature predictions than the previous report (AR4) and has even increased the range of uncertainty for climate sensitivity (see item 7 below) yet it has increased its claimed level of certainty of future higher temperature-rise predictions over that claimed for previous reports in direct contradiction of its own results.
5. IPCC’s temperature prediction for this century alone exceeds the level that can be achieved over any time period even if the world’s fossil-fuel reserve is completely consumed in that time. (see Table 4 and Figure 18). Maximum temperature increase stays below 1o C.
6. The claim that we have a limited time for action on climate (i.e., that we will “lock in” a future temperature increase without immediate action) is false. As soon as the input to atmospheric CO2 is decreased (as, for example, by running out of fossil fuel) temperature decreases along with it no matter what CO2 level and temperature are reached in any given time frame (see Figure 18).
7. IPCC has made an issue over a parameter called Climate Sensitivity (CS), which is defined either by the temperature increase from doubling atmospheric CO2, or the temperature increase per unit of induced climate forcing function, (w/m2)-1 across the atmosphere. It has no intrinsic utility, but has had an impossibly high range of values ascribed to it, apparently for use in magnifying the true warming capability of CO2 (see CALCULATIONS: Climate Sensitivity).
8. The current popular hypothesis among supporters of global warming, that the recent hiatus in surface warming is due to an increase in the fraction of heat that is being stored in the deep ocean and that “the heat will come back to haunt us sooner or later,” is totally unsupported by the laws of physics (see CALCULATIONS: Ocean Stored Heat ).
9. The maximum possible achievable CO2 is far below the level of harm to animals and as well as well below the optimum level for plant growth and health. (See CALCULATIONS: Effect of CO2 Levels).
10. Over-predictions of achieved temperature increases published with the approval or the sponsorship the IPCC are as pervasive as they are large. If these were unbiased calculations there should be approximately as many under-predictions as over-predictions.
Both data and analysis strongly contradict the claims being made for alarming global warming caused by CO2. The magnitude and prevalence of the over-prediction errors are sufficient to impugn the objectivity of the claimants.
A limit analysis means that instead of attempting to accurately model all the factors involved in calculating CO2-caused temperature increase, a model is generated that can be accurately calculated and that can be reasonably assured not to underestimate the true temperature rise. Results of such studies are useful for policy making. The following limits apply for the calculations of this analysis.
1. The absolute maximum increase in temperature is based on permanently inserting all of the world’s fossil carbon into the atmosphere. That effect can be calculated accurately with the available tools, and few would deny that it is an extreme upper limit to what could actually happen in any time frame. Yet that temperature rise is only one degree Celsius more than the minimum of the range and two degrees Celsius less than the maximum of the range predicted for this century, alone, by IPCC contributors (see Table
4). But, instead of stopping with that extreme upper limit, this analysis backs down from that to an increase which is still a credible upper limit by allowing more and more of the biosphere to participate in the receipt of atmospheric carbon to the point where there is no more decrease in the maximum computed atmospheric temperature, and that is used for the temperature limit of what CO2 increase can ever achieve. That point is reached before the complete biosphere (which means all of the atmosphere, land and ocean) are included among the recipients. It is reached by allowing all of the atmosphere and land plus only about the top 15% of the ocean to receive atmospheric carbon. Its value is 1oC.
2. Heat transfer out of the atmosphere is assumed to be limited to outer space. No credit is taken for heat transfer to the rest of the world (land and ocean) which actually occurs and cools the atmosphere because of its vast heat capacity. The assumption avoids complexity and the reliance on questionable parameters while assuring a conservative value of atmospheric temperature rise. (See Reference 9).
3. The equation of average temperature increase in the atmosphere derived in Reference 10 is based on the maximum temperature in the atmosphere (that at the earth’s surface) instead of an aggregate average temperature appropriate for all radiant transfers to outer space from the atmosphere . Such a choice ensures a conservative estimate. The equation is:
DT = [(HA/HB)1/4-1]*TB (2)
-subscripts A and B refer to conditions after and before addition of CO2,
- H is the total heat rate to the atmosphere, and
- T is the atmospheric temperature
4. This analysis claims no negative feedback from CO2 creation of water vapor. The heat of vaporization required to create it is a negative input but is approximately replaced when the vapor condenses at higher altitudes and to a lesser extent by the ghg heating effect of water vapor.
5. The carbon input to the atmosphere is assumed to be larger (at 10 petagrams per year) than any so far recorded, all from fossil-fuel and continuing at that maximum rate until world fossil carbon is consumed. The world fossil carbon level used is highest of the carbon balances listed.
A limit analysis cannot be divorced from human judgment. But neither can the complex models of the IPCC (general circulation models, GCMs) which rely on a great number of poorly known parameters. The climate is too poorly understood to be analyzed with extreme precision. Limit analysis is a valid check on attempts to model all contributing factors in the IPCC analyses. In this regard Table 3 shows the results of using even more conservative assumptions. The highest of these results barely reaches the lower limit of the IPCC temperature prediction just for this century.
Water Vapor Effect
Not all of the water in the atmosphere can be attributed to feedback from greenhouse gases. Most of it got there by other mechanisms. It is appropriate to separate such water vapor (called “existing” water vapor) from that which is produced specifically by CO2 addition (feedback). The effects of water vapor and clouds on CO2’s temperature increase are shown in Figures 5, 6 and 7.
Clouds have a competing effect on temperature. They absorb IR to add heat to the atmosphere but the white tops reflect the sun’s energy to cool it. The reflection dominates, but there is a single exception in the sub-visual cirrus clouds of Figure 6 which has a small positive effect,
*This climate region and weather conditions are noted in Appendix B to duplicate average world climate conditions
as shown by the figure, since their atmospheric temperature rise slightly exceeds the clear-sky level because their reflection of incoming light is limited. Such sub-visual cirrus clouds are so rare (11) that their average effect is negligible. The average net effect of increasing water in any phase (vapor, liquid or solid) and by any means is one of cooling. Figure 5 and 6 results are calculated with Modtran (Appendix B) and Equation 2.
Figure 7 shows an undeniable negative correlation between extent of world cloud cover and world temperature.
Figure 7. 26-Year Record of World Temperature vs. Cloud Cover (12)
Changes in ocean surface temperature that are caused by ghg heating from water vapor can produce more water vapor for a potential positive feedback. But there is an approximate net zero effect from the heat drawn from the air to vaporize the water because that heat is returned when the vapor condenses (cloud formation). Of course, cloud formation has a net cooling effect as shown in Figures 6 and 7. Any direct feedback from ghg-formation of water vapor is appropriately ignored. The vapor produced by the CO2-produced ghg heat is calculated by the method developed by Wolff (13) that transmits ghg produced heat in the atmosphere into the ocean. Tabulated results of the Clausius-Clapeyron Equation (Table 2 of Reference 10) or similar relations are used to determine resulting water vapor for the Modtran input for iterations between heat-produced vapor and vapor-produced heat. Vapor pressure of water is dependent only on its temperature. Temperature is virtually constant (=earth surface temp) in the epipelagic zone of Fig. 8. Greater assumed depth has little effect on vapor pressure.
Figure 8. Typical Ocean Temperature Profile (14)
Saturation is the diminishing effectiveness of CO2 absorption of IR with its increased concentration. The absorption of light (including infrared) at a given wavelength λ is governed by Beer’s Law, which states that absorption (A) of light through a component in a mixture decreases exponentially with the path length (l) and concentration (c) of the component, as expressed by the equation
Aλ = 1 – exp -(kλ * c * l), (3)
where kλ is a constant specific to the component and wavelength.
Although the constant kλ for CO2 in air varies greatly throughout the infrared range, the influence of Beer’s Law can be seen by analysis with programs such as Modtran (Appendix B). The decreasing slopes of the curves in Figures 5 and 6 are the results of Beer’s Law (saturation). The current atmospheric CO2 is already well saturated. There is no argument against the negativity or the magnitude of this feedback. The feedback graphs of Figures 1 and 2 (which are basically of saturation) are calculated by Modtran, not by Equation 3.
Feedback from Out-gassing CO2 from a Warmed Ocean
A warmed ocean releases CO2 through Henry’s law (15) as depicted graphically in Figure 9.
A limit analysis that allows all atmospheric ghg heat from CO2 to be deposited in the ocean shows the insignificance of this feedback. In the vicinity of average ocean temperature (15o C) Figure 9 shows carbon dioxide solubility decreases by 0.085 g of CO2 per kg of water per deg. C of water temperature increase (the ocean’s greatest rate of decrease). A one-degree change of a kg of water is a kilogram calorie. The weight of a 1 m2 column of air at the earth’s surface is 10,357 kg. A Modtran derived forcing function across the atmosphere plus equation 2 shows a temperature increase to the atmosphere of approximately 0.75 oC for doubling from 400-to-800 ppm, or increasing its heat by 7768 calories. Conservatively assuming all of that heat is transferred to the ocean produces 470 grams of CO2 in the meter-squared atmosphere column using 71 percent of atmosphere transferring to the ocean. 800 ppm of CO2 in the atmosphere weighs 125,000 grams per square meter on the surface. A 470-gram increase above 125,000 is the same fraction as 3 parts per million (ppm) above 800. With the extreme conservatisms built into the analysis, this value is truly negligible.
Feedback from removal (leakage) of CO2 from atmosphere
Feedback from water and from saturation is determined within the Modtran code because the results are dependent upon the input of water-vapor and CO2 levels and climate conditions chosen. That from CO2 removal must be determined separately to govern the CO2 level chosen for input to Modtran. The negative removal feedback is significant, generally ignored and probably the least studied. The reason it is significant is that the atmosphere has but a small fraction of the carbon in the earth’s biosphere, about 2 percent, as illustrated in the three world carbon balances of Appendix A. It exchanges its carbon with the remainder of the biosphere at more than twenty times the rate that man’s fossil-fuel combustion has ever been able to put carbon into it. The figures in Appendix A show that carbon dioxide molecules cross the biosphere boundaries in both directions. All CO2 from fossil-fuel combustion goes into the atmosphere, but the atmosphere ultimately keeps but a small fraction of it.
The time-dependent solution for the CO2 content of the biosphere reservoirs is required to determine the removal of CO2 from the atmosphere. The net transfer across the boundary between adjacent reservoirs A and B from A to B is the expulsion rate from A minus that from B. A key assumption in the solution is that the expulsion rate out of any boundary of any reservoir is proportional to the content of that reservoir which requires that there is a time-invariant mean probability that a molecule will escape through any boundary in a unit time. That is true if the relative spatial distribution within reservoir is invariant with time and that requires short transit times across the dimensions of the reservoir. The land reservoir is modeled as a surface phenomenon, with vertical transmission which assures 0 transit time. In the atmosphere the average Maxwellian velocity of the molecules is roughly 300 m/sec allowing a path traversal equal to roughly 15 atmospheric depths in an hour which would explain the constant relative concentration (parts per million – ppm) of CO2 with altitude. The very short horizontal transit times are verified by the Keeling measurements of atmospheric CO2 at Mauna Loa which display semi-annual plant-produced variations in CO2 level essentially as they occur, but Mauna Loa is half the Pacific Ocean removed from the world’s major plant activity.
Only two biosphere boundaries are included: the land-atmosphere and the ocean-atmosphere. Any direct transfer between the land and ocean is ignored. CO2 is the predominant chemical form of carbon that transfers it across these boundaries. All other forms are ignored here. The differential equations governing the time-dependent CO2 content of the biosphere reservoirs are solved numerically by finite difference equations with selected time increments. The finite time increments of the integration place another condition on transit times across the individual reservoirs. If the transit time exceeds the integration-time increment, because the equations calculate complete filling of each reservoir during the increment, the true rate of CO2 spread throughout the reservoir would be inappropriately amplified. Transit times for the land and atmosphere do not violate such a condition, but it can be met for the ocean only by restricting the size of the individual ocean reservoirs which means they cannot exceed 200 pg, which is their transfer rate per yearly time increment. This restriction was not employed in Reference 10, thereby overestimating the rate of transfer to the deep ocean and underestimating the maximum level attainable in the atmosphere
Transit and distribution of ocean carbon
CO2 dissolved in the ocean travels with the ocean water which is glacially slow compared to atmospheric molecular velocities. When atmospheric CO2 is absorbed by the ocean, only 0.6% remains as dissolved molecular CO2, the remainder is converted to bicarbonate ions and much less to carbonate ions. All its forms are subject to transport by a variety of vertical and horizontal ocean currents as well as by diffusion and simply by falling to the ocean floor (as dead phyto- and zoo-plankton). A model which captures all these effects would be impractical even if they were well known. But it is possible to define and calculate defensible bounding limits that allow an approximation of the associated time and space distribution of the carbon emitted from the atmosphere. The dissolved CO2 is governed by Henry’s Law (15)which expresses the relationship between the partial pressure, P, of atmospheric CO2 and its concentration, C, in moles per liter in the ocean as dissolved in the ocean.
P = KC. (3)
where K is the temperature-dependent Henry’s Law constant. Its temperature dependence is sufficient that arctic waters can dissolve more than twice that of tropical waters which produces a net exchange in different parts of the ocean-atmosphere boundary that is in the opposite direction at the same time.
The Woods Hole model of Appendix A is the basis for the time studies presented here because it is the most conservative of the three—having the largest fossil carbon reservoir and greatest input rate to the atmosphere.
Its total biosphere has 43,620 petagrams (pg) of carbon and comparison of the three balances shows the exchange between ocean and atmosphere and that between the shallow ocean and the deep ocean to be approximately 200 pg per year. That determines a maximum size of 200 pg for the individual ocean reservoirs modeled. That transfer rate is assumed to be maintained throughout the ocean. Fortunately, Microsoft’s EXCEL spreadsheet has the capability to solve over 200 simultaneous difference equations over a time span of thousands of years. (See Figures 12 through 17). These figures demonstrate the great time delay for carbon exiting the atmosphere to reach deep ocean regions.
No information of the physical dimensions of the multiple ocean regions (segments) used to model this time delay is implied in the calculations or graphs. They are based on identical 0-time CO2-content of the regions. But the atmospheric carbon does have to traverse all segments above a given segment to reach it. This requirement is approximate since vertical ocean currents can cause the CO2 to bypass segments. Ignoring such bypass should be conservative because that will require longer times to move CO2 away from the atmosphere, delaying its concentration decline. Those inputs of man-made CO2 that start into the atmosphere beginning at time 0 and end at the depletion time of fossil fuels are the ones modeled. The differential equations associated with a single-segment ocean model are (from reference 10):
A’ (t) = Fa(t)*F(t) + La(t)*L(t) + Ca(t)*C(t) +A(t)*[1-Al(t)-Ac(t))]
L’(t) = -La(t)*L(t) +Al(t)(t)*A(t)
C’(t) = Ac(t)*A(t) –Ca(t)*C(t)
F’(t) = -Fa(t)*F(t) (2)
Where ‘ indicates differentiation with respect to time, t. As indicated, all terms are functions of time, t. A, L, C, and F are the CO2 quantities in the atmosphere, land, ocean and fossil fuels, respectively. The time-dependent transfer rate from atmosphere to ocean is Ac; Ca is from ocean to atmosphere, etc. In the equations used the fossil-fuel equation is eliminated and fossil-fuel input is simply modeled as a constant input, F, to the atmosphere which results in the following difference equations to be solved numerically. The associated entries for each year n+1 in terms of the entries for year n follow as Equations 3. For year 0, the current values of the carbon content and transfer rates are used and the ocean region modeled is the 800-petagram (pg) shallow ocean. The difference equations for the biosphere modeled as the atmosphere, land and four equal segments of the shallow ocean are listed below.
An+1 = An(1- 0.244) + 0.05Ln + 0.5C1n +F
Ln+1 = Ln(1-0.05) + 0.122An;
C1n+1 = 0.122An + 0.5C2.n
C2n+1 = 0.5C1.n + 0.5C3.n
C3n+1 = 0.5C2.n + 0.5C4.n
C4n+1 = 0.5C3.n - 0.5C4.n (3)
Table 1 shows the results of this integration for ten years at a uniform rate of 10 pg/year and Figure 10 graphs the results of for expanding into all regions in which the CO2 is allowed to escape. Fractional increase is the ratio of CO2 content to that at time-0 minus one, and is plotted as the ordinate of the CO2-content curves because it displays the proper perspective of increases as positive and decreases as negative. The escape of CO2 into these regions cannot actually be restricted as indicated, but the modeling of it illustrates the relative importance of the various regions in atmospheric CO2 removal.
Figure 10 implies that the slow rate of transfer into the ocean does not allow a sufficiently rapid escape of CO2 from the atmosphere below about 12 % of the ocean’s initial carbon content to affect (decrease) the maximum level in the atmosphere until after the input to the atmosphere ceases. The huge capacity of the deep ocean contributes almost nothing to the limitation of the atmosphere’s maximum level.
Table 1. Carbon level in Biosphere Regions from 10-pg/year Input*
|Year||AtmosphereCO2||Land/PlantCO2||Ocean 1CO2||Ocean 2CO2||Ocean 3CO2||Ocean 4CO2|
*limited to atmosphere, land and shallow ocean in 4 ocean regions
In Figure 10 the atmosphere and its combination with the first three levels below it are small enough in capacity that its maximum level achieved is minimally dependent on either the rate of its CO2 input or the number of segments modeled in the receiving volume. But with greater depths of ocean included, CO2 has to travel distances that are far enough removed from the atmosphere that its removal is impeded not by presence of an artificial barrier but by the rate which CO2 can travel through the ocean thereby impeding its rate of escape. The size of time increments and segments modeled both can both influence carbon transit before the barriers are effective. Restriction to the atmosphere, only, allows no removal, of course. The ocean regions are all modeled with 200 pg segments. Figure 11 shows the temperature effects associated with carbon levels of Figure 10.
The slow rate of ocean travel explains why the maximum level in the atmosphere is constant from about 15-to-100 percent of the total ocean being included in the receiving volume. Once the input to the atmosphere ceases, the greater ocean volumes eventually remove increasingly more CO2 from the atmosphere and the ultimate fractional increase is only 0.23 with the total ocean involved versus the 12.2 there would be if there was no removal. As indicated in Figures 12 through 17, the time required is several millennia. Figure 11 shows the temperature increases associated with the carbon levels of Figure 10. All these are hypothetical except for the final one, the land plus the entire ocean, because the regions cannot be physically excluded from receiving atmospheric carbon. Table 2 summarizes temperature results from the CO2 levels plotted in Figure 11. Comparison of Figures 12 and 13 illustrates the small difference in maximum atmospheric level achieved when modeled with and without an artificial barrier at its lower extremity.
Table 2. Achievable Atmospheric Temperature Increases from Fossil-Fuel
Carbon in the Biosphere with 10 pg/year input for 1000 years
|Description ofReceiving region*||BiosphereFraction||MaximumDT, OC||Stable DT,OC **||DT for thisCentury, oC|
|A + 2% ocean||0.037||1.87||1.87||0.40|
|A + L||0.065||1.45||1.45||0.28|
|A + L +2%||0.085||1.32||1.32||0.26|
|A + L + 8%||0.145||1.05||0.99||0.24|
|A + L +12%||0.185||1.00||0.82||0.23|
|A + L + 25%||0.315||0.98||0.51||0.22|
|A +L + 50%||0.565||0.98||0.32||0.21|
|A + L + 100%||1.0||0.98||0.20||0.21|
* A is for atmosphere, L for land and the number is percent of ocean.
** When the biosphere reaches equilibrium following CO2 cessation.
The difference totally disappears beyond the 12% fraction of ocean involvement and the maximum fractional increase in the atmosphere never drops below 1.57 with this model as shown by the overlaying of these plots.
Figure 14 shows the results of limiting the receiving volume of the ocean to 25%. Figure 15 shows the distribution for limiting down to the top 50% of the ocean. Figure 16 show representative levels with the entire ocean included the biosphere receiving volume.
Figure 17 provides added perspective that may be obscured with 204 separate plots. “Ocean 10” in the legend refers to the 10th ocean segment below the surface, etc. There are 204 such segments of 200-pg of initial carbon each in the total ocean. The fact that modeling 25%, 50% and 100% of the ocean as part of the receiving biosphere produces identical maximum fractional increase in atmospheric level indicates that only the upper regions of the ocean contribute to draining the excess CO2 from the atmosphere in times less than 1000 years (while CO2 is still being inserted in the atmosphere). In fact, some of the lower segments of the ocean are unaffected by atmospheric increases for 2000 years after the start of atmospheric input (0 time) and 1000 years after the atmospheric level is modeled as having reached its maximum.
The curve labeled “all biosphere” matches the ratio of inserted carbon to total initial biosphere carbon. It is, of course, linear with time and reaches a constant maximum value in 1000 years, the assumed duration of carbon insertion. The all-biosphere maximum is inversely proportional to the total biosphere modeled and all segments match that (constant) value for their ultimate fractional increase. For the total world fossil carbon reserves (10,000 pg) and the total biosphere carbon (43,620 pg) this fraction is 0.23.
Maximum time distribution of CO2 and temperature
Figure 18 shows the temperature response to the atmospheric increase in
CO2. Any delay in response is obviously negligible. The claim that we can “lock in” a response for the future by currently adding CO2 is not valid. Figure 18 also demonstrates the effect of saturation because the disparity between relative CO2 increase and relative temperature increase is proportional to the increase in CO2.
Deep Ocean Storage of Heat
Check on Atmospheric CO2 Exchange
Appendix A shows typical world carbon balances from Reference 22. There are many of these in Reference 22 and they all show approximately 200 pg/year being exchanged between the atmosphere and its surroundings, about equally between the land and the ocean at 100 pg each. IPCC’s 4th assessment report notes that they are “estimates” without providing any more detail than that. Because of their wide acceptance, the approximate average of these exchange rates was chosen for this analysis. However, the atmospheric loss rate of carbon-14 from the atmosphere that was deposited there by the atmospheric bomb tests of the 1950’s and 1960’s indicates the relaxation time for carbon’s removal from the atmosphere is 14 years as indicated in Figure 19 from beyond 1970 (16), which would determine an exchange rate with its neighbors of approximately 50 pg/year. This much lower exchange rate delays the escape of CO2 from the atmosphere to the extent that the all-time maximum temperature increase is 1.32 C instead of 0.98 C, still lower than the minimum of the IPCC predicted temperature rise for this century, alone. Figure 20 compares the atmosphere temperature histories between the 50-pg/year and 200-pg/year exchange rate.
Figure 19. Decline of Carbon 14 Following Bomb Tests (16)
Ocean Stored Heat
A response from Kevin Trenberth, noted contributor to published IPCC reports, to the IPCC’s 21st-century over-prediction of temperature rise (see Figure 4) has been that the deep ocean has started storing additional heat and that the extra heat “will come back to haunt us sooner or later” (17). The problem with that conjecture is that it violates the fundamental laws of thermodynamics. The heat to the ocean comes from above, and extra heat cannot be transferred downward without increasing the temperature gradient between the surface and the deep layers. Any such increase will decrease the lower temperature relative to the upper temperature, contrary to the Trenberth hypothesis. The temperature dependence of water density gives a boost to thermodynamics in this refutation. Warm water rises and cool water sinks. Trenberth’s hypothesis cannot be valid.
Increasing Limits of the Limit Analysis
The specific limits chosen for the limit analysis were based on confidence that the calculational parameters were adequately conservative. These have been tested by doubling the insertion rate chosen, by doubling the total amount inserted, and by doubling both. The results are shown in Table 3. None of these produce all-time high temperatures that even approach the IPCC’s temperature projections for only as long as this century
Table 3. Temperature Effects of Insertion Rate and Total Carbon Inserted
|10,000 pg*||10 pg/year*||0.98||0.19||0.21|
|10,000 pg||20 pg/year||1.09||0.19||0.37|
|20,000 pg||10 pg/year||1.29||0.35||0.21|
|20,000 pg||20 pg/year||1.41||0.35||0.37|
*Limit basis for the results of this analysis.
**Time of maximum is the ratio of total insertion to rate of insertion
Comparison with IPCC Values.
Table 4 values are the summary values from the results of the latest IPCC Report, AR5 (18). It contains the statement that most predictions fall between 1.5 and 4 degrees Celsius temperature rise for the 21st century and this is the basis for comparison. The report also refers the results back to “the period between 1850 and 1900” without specifying a CO2 level or temperature of that period. Neither the CO2 levels nor temperature levels for that period are well known. However, the CO2 levels and temperature estimations of that period do permit an approximate comparison. Therefore, the best comparison that can be made for 21st century temperature rise is an approximate one between two different approximations—the IPCC models and this one.
There is a general consensus of a temperature rise of 0.7-0.8 deg. Celsius since 1850. Therefore the temperature records of Table 4 derived from the AR5 summary for policy makers, SPM (17), are reduced by 0.8 to compare with the temperature rise predicted here in Table 3 for the 21st century. Alternatively, the rise in temperature predicted by the method of this article in going from the 280 ppm of 1850 to 400 ppm used for the starting level for this study can be added to the values in Table 3 to compare with the value listed in AR5 from 1850. This value is 0.39. Another comparison is by extrapolating to the end of this century the results of Figure 4 of the IPCC predictions reported by Christy and Spencer and those by using the method of this study. The IPCC does not publish the results of studies such as those used by Christy and Spencer. These are presumed to be obtained from the analysts who produced them for the IPCC.
Table 4. Comparative Values for Temperature Rise
|Method||Calendar Range||Range of Prediction oC|
|IPCC||1850-2100||1.5 – 4|
|This Study||1850-2100||0.6 – 0.76|
|IPCC||2000-2100||0.7 – 3.2|
|This Study||2000-2100||0.28 – 0.44|
|Figure 4 IPCC*||2000-2100||2.7 (single value)|
|Figure 4 This Study*||2000-2100||0.32 (single value)|
*Figure 4 curve extrapolated to end of this century
The three different sets of comparison in Table 4 show that IPCC results exceed those of this study by factors ranging from 2.5 to 8.4. Recall that Table 3 which shows that all of the conservatively chosen CO2 reservoir levels and CO2 insertion rates predict all-time maximum levels that fall below the lowest IPCC prediction level (1.5 C) for just this century.
A final metric of comparison is “climate sensitivity,” CS, defined both as the degrees Celsius of temperature increase from doubling atmospheric CO2 or as the increase per unit of forcing function, (w/m2)-1 (19). It is dependent on the level from which it is doubled as well as the climate conditions during its doubling. All IPCC reports have listed a range of values for it, which have been the same, from l.5 to 4.5 C, for doubling except for AR4 which raised the lower limit to 2, but that was again reduced to 1.5 C for AR 5. Because of its dependencies, listing a range is necessary, but the listed range is very high. However, none of the IPPC’s means of determination are able to isolate the heating due just to carbon dioxide so its utility seems very limited; and there is controversy over what time frame should be involved. The only way to isolate the CO2 effect is to use a code like Modtran with Equation 2 at a given input CO2 and then doubling that keeping other conditions constant to get the temperature-rise difference between the two. Using a broad scope of choices of climate and weather conditions available in Modtran, no condition was found where it exceeded a value of 1.0 C (fifty percent lower than the lower limit of the ranges listed in the IPCC reports). The average is about 0.7. Reference 9 calculates a value of 1. Its only obvious use would appear to be as another means of exaggerating CO2 effect.
Effect of CO2 Levels on animals and plants
Reference 20 indicates that human beings can function well at levels up to 5000 ppm of CO2 (12.5 times the current atmospheric level) without impairment and that this is a common level in submarines. The systemic effect above that level is caused by oxygen dilution. Reference 21 indicates optimum level for plants (most efficient photosynthesis) is 1000 ppm. A level as high as 1000 would seem desirable. As long as the level does not approach 5000, there should be no concern.
APPENDIX A. ATMOSPHERIC CARBON CYCLES.
Figure A1. IPPC World Carbon Cycle
. Cycles depicted in this Appendix are from Reference 21 and are typical of the many that are shown. All the depicted cycles are similar but show variation in the flows between reservoirs as well as the reservoir contents
Most show two oceans (surface and deep ocean) and indicate a connection between the two but not all, as in A3, show the rate of flow between these. Some combine these oceans into one, but the number of ocean regions modeled does affect the atmospheric temperature rise from added carbon
The model choice for this analysis is that which maximizes both the carbon content in the atmosphere and fossil fuels as well as the rate of fossil-carbon input to the atmosphere which is that of Figure A3.
Figure A2. University of New Hampshire World Carbon Cycle
Figure A3. Woods Hole World Carbon Cycle
APPENDIX B. USING MODTRAN (Provided by Calvin M. Wolff)
MODTRAN (MODerate resolution atmospheric TRANsmission) is a computer program designed to model atmospheric propagation of electromagnetic radiation for the 100- 50,000 cm-1 (0.2 to 100 um) spectral range.
The most recently released version of the code, MODTRAN5, provides a spectral resolution of 0.2 cm-1 using its 0.1 cm-1 band model algorithm.
Some aspects of MODTRAN are patented by Spectral Sciences Inc. and the US Air Force, who have shared development responsibility for the code and related radiation transfer science collaboratively since 1987. The acronym MODTRAN was registered as a trademark of the US Government, represented by the US Air Force, in 2008.
All MODTRAN code development and maintenance is currently performed by Spectral Sciences while the Air Force handles code validation and verification. Software sublicenses are issued by Spectral Sciences Inc., while single-user licenses are administered through Spectral Sciences’ distributor, Ontar Corporation.
MODTRAN5 is written entirely in FORTRAN. It is operated using a formatted input file. Third parties, including Ontar, have developed graphical user interfaces to MODTRAN in order to facilitate user interaction and ease of use.
MODTRAN is accessible to the public at http://forecast.uchicago.edu/Projects/Modtran.html.
When you access the url above, a menu will appear, as follows:
Modtran IR in the Atmosphere
Where Iout is the infrared heat radiated outward from the earth at 70 km altitude.
On the graph, the smooth lines represent perfect blackbody radiation at the temperatures cited in the legend on the graph. The red, jagged line is the earth’s actual infrared emission outward at 70 km altitude. The horizontal axis is in units of wavenumber, proportional to frequency and inversely proportional to wavelength. To convert wavenumbers to wavelength in microns, simply divide the wavenumber value into 10,000; i.e., 10,000 wavenumbers corresponds to a wavelength of 1 micron. The visible spectrum is from 0.8 to 0.4 microns.
Please note that the result is for the tropical latitudes, no clouds or rain, with the instrument or observer looking down to the earth.
To run simulations for the average earth, set “Locality” to “1976 U. S. Standard Atmosphere” and change “No clouds or rain” to “NOAA Cirrus Model (LOWTRAN 6 Model)”.
When you simulate at these conditions, you will see that the ground temperature changes from 299.7K to 288.2K, corresponding to the 15C that is usually taken as earth’s average surface temperature. The radiation emitted from earth, Io is 242.782 w/m^2.
To compare the heat loss from earth at various CO2 levels, use the 1976… and NOAA…. settings, leave all the rest the same, and set the CO2 ppm to 390, which is closer to the current amount. Record Io (watts/m^2) for that simulation. Then change increase the CO2 amount to whatever you choose. Doubling atmospheric CO2 would be 2 x 390 ppm = 780 ppm. When you double CO2 (780 ppm), you will see that the new Io (heat lost to atmosphere) drops to 240.336. Therefore, using Modtran, the heat loss from the earth by doubling CO2 is 242.782 – 240.336 = 2.446 w/m^2, which is the greenhouse effect of doubling CO2 (an estimate).
To study (estimate) the effects of changes in other atmospheric constituents (CH4, Ozone & water vapor) at any given, constant CO2 ppm, do as follows: multiply the default quantities (17 ppm CH4, 28 ppb O3, water vapor scale) by the 1 + the amount you want to change them. If you want a 20% increase, multiply by 1.2.
Modtran gives a good, but not the best estimate of radiative heat loss. Other programs, like SpectralCalcTM should give more accurate estimates. Modtran averages over the entire earth over an entire year.
3. Remote Sensing ISSN 2072-4292, http://www.mdpi.com/journal/remotesensing;
Article: On the Misdiagnosis of Surface Temperature Feedbacks from
Variations in Earth’s Radiant Energy Balance, Roy W. Spencer and William D. Braswell, ESSC-UAH, University of Alabama Huntsville
4. Dr. Roy Spencer, Global Warming Home/Blog. Global Warming Slowdown: The View from Space, April 16, 2013.
http://data.giss.nasa.gov/gistemp/tabledata_v3/GLB.Ts+dSST.txt; linearized by C. Bruce Richardson.
7. The IPCC Under the Microscope, http://mclean.ch/climate/IPCC.html
9. Stephen E. Schwartz, “Heat Capacity, Time Constant and Sensitivity of Earth’s Climate System,” Brookhaven National Laboratory, June, 2007.
11. http://www.climate4you.com/ , click on Clouds and Rain
12. The International Satellite Cloud Climatology Project and University of East Anglia‘s Climatic Research Unit. Last cloud data used: December 2009. Last figure update: 4 September 2011.
14. Internet Search with subject: “NWS Jet Stream Online School for Weather” Table entry; “Ocean Layers”
15 Engineering Tool Box, Solubility of gases in water: http://www.engineeringtoolbox.com/gases-solubility-water-d_1148.html
16. Gösta Pettersson, “The bombtest curve and its implications for atmospheric carbon dioxide residency time,” “Watts up with That,”July 1, 2013
17. Kevin Trenberth and John Fasullo, Title: “Tracking Earth’s Energy,” Science, April 16, 2010.
20. Human exposure to CO2 http://www.blm.gov/pgdata/etc/medialib/blm/wy/information/NEPA/cfodocs/howell.Par.2800.Fi
21. Optimum plant benefits from CO2
22. Internet search with subject: “Images for global carbon budget”
The author gratefully acknowledges the following contributions to this effort.
To Calvin M. Wolff for many useful technical interactions, for critiques and corrections, for providing useful methodology and references, and for contributing to its writing
To Neil Brown for much encouragement and for useful reviews of several drafts of the document.
To Ed Berry for early advice and encouragement and for posting prior versions on his website for comments.
To William Happer for a critical review of the previous draft, for helpful suggestions, for encouragement and for lucid explanation of the physical processes involved.
To Arthur Goldman for many reviews and helpful suggestions
About the Author
Bryce Johnson is retired professional nuclear engineer in the state of California with a 45-year career in nuclear-reactor and nuclear-weapons research. His education includes BS (ME), University of Idaho; MS (NE), North Carolina State University and PhD (ME), Stanford University.