tag:blogger.com,1999:blog-16397380905451389332017-11-22T10:07:13.452-05:00Home Climate AnalysisInvestigation of climate and weather by an engineer. See our <a href="http://homeclimateanalysis.blogspot.com/p/status-report.html">Summary to Date</a> page for the story so far.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.comBlogger173125tag:blogger.com,1999:blog-1639738090545138933.post-71448486171592384442016-10-09T21:15:00.004-04:002016-11-08T11:03:31.628-05:00Falsification of Anthropogenic Global WarmingThe theory of Anthropogenic Global Warming, so far as we understand it, consists of the following two assertions.<br /><br />(1) If we increase the concentration of CO2 in the atmosphere to 600 ppmv, we will cause the world to warm up by at least 2°C. (The concentration in pre-industrial times was 300 ppmv and is currently 400 ppmv, where ppmv is parts per million by volume.) <br /><br />(2) If we continue burning fossil fuels at our current rate, emitting 10 petagrams of carbon into the atmosphere every year, we will raise the concentration of CO2 in the atmosphere to 600 ppmv within the next one hundred years.<br /><br />We can falsify the second assertion using our observations of the carbon-14. We present a detailed analysis of atmospheric carbon-14 in a series of posts starting with <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">Carbon-14: Origins and Reservoir</a>. Here we present a summary, with approximate numerical values that are easy to remember.<br /><br />Each year, cosmic rays <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">create</a> 8 kg of carbon-14 in the upper atmosphere. If carbon-14 were a stable atom, all carbon in the Earth's atmosphere would be carbon-14. But carbon-14 is not stable. One in eight thousand carbon-14 atoms decays each year. The rate at which the Earth's inventory of carbon-14 decays must be equal to the rate at which it is created. There must be 64,000 kg of carbon-14 on Earth.<br /><br />The Earth's atmosphere <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">contains</a> 800 Pg of carbon (1 Pg = 1 Petagram = 10<sup>12</sup> kg) bound up in gaseous CO2. One part per trillion of this carbon is carbon-14 (1 ppt = 1 part in 10<sup>12</sup>). There are 800 kg of carbon-14 in the atmosphere. That leaves 63,200 kg of the total inventory somewhere else. We'll call this "somewhere else" the carbon-14 <i>reservoir</i>.<br /><br />Each year, 8 kg of carbon-14 is created in the atmosphere by cosmic rays, and each year the atmosphere <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">loses</a> 8 kg of carbon-14 to the reservoir. (Here we are ignoring the 0.1 kg of atmospheric carbon-14 that decays each year.) There is no chemical reaction that can separate carbon-14 from normal carbon. Every 1 kg of carbon-14 that leaves the atmosphere for the reservoir will be accompanied by 1 Pg of normal carbon. <br /><br />Consider the atmosphere before we began to add 10 Pg of carbon to it each year. The mass of carbon in the atmosphere is constant. If 1 Pg of carbon leaves the atmosphere and enters the reservoir, 1 Pg of carbon must go in the opposite direction, leaving the reservoir and entering the atmosphere.<br /><br />The only way for there to be a net loss of carbon-14 from the atmosphere to the reservoir is if the concentration of carbon-14 in the reservoir is lower than in the atmosphere. The only place on Earth that is capable of acting as the reservoir is <a href="http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-reservoir-is-ocean.html">the deep ocean</a>, in which the concentration of carbon-14 is 80% of the concentration in the atmosphere. Each year 40 Pg of carbon leaves the atmosphere and enters the deep ocean, carrying with it 40 kg of carbon-14, while 40 Pg of carbon leaves the ocean and enters the atmosphere, carrying with it 32 kg of carbon-14. The result is a net flow of 8 kg/yr of carbon-14 into the ocean. Furthermore, the ocean contains 63,200 kg of carbon-14 in concentration 0.8 ppt, so the total mass of carbon in the oceans is roughly 80,000 Pg.<br /><br />With the ocean and the atmosphere in equilibrium, 40 Pg of carbon is absorbed by the ocean each year, and 40 kg is released by the ocean. If we were to double the quantity of carbon in the atmosphere, we <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-14-probability-of-exchange.html">would double</a> the amount absorbed by the ocean each year. Instead of 40 Pg being absorbed each year, 80 Pg would be absorbed. We could double the concentration of carbon in the atmosphere by emitting 40 Pg/yr. But we emit only 10 Pg/yr. Our emissions are sufficient to increase the mass of carbon in the atmosphere by 25%, after which everything we emit will be absorbed by the oceans. The oceans contain 80,000 Pg of carbon. If we add 10 Pg/yr, it will take roughly <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-with-ten-petagrams-per-year.html">eight thousand years</a> to double the carbon concentration in the oceans, after which the concentration in the atmosphere will double also.<br /><br />Back in the 1960s, atmospheric nuclear bomb tests doubled the concentration of carbon-14 in the atmosphere. Such tests stopped in 1967. In our more precise calculation <a href="http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-bomb-tests.html">we predict</a> that the concentration of carbon-14 must relax after 1967 with a time constant of 17 years, so that it would be 1.37 ppt in 1984 and 1.05 ppt in 2018. The concentration did relax afterwards, with a time constant of roughly 15 years, and in 2016, the carbon-14 concentration in the atmosphere is indistinguishable from its value before the bomb tests. During that time, almost every CO2 molecule that existed in the atmosphere in 1967 passed into the ocean and was replaced by another from the ocean. Anyone claiming that our carbon emissions will remain in the atmosphere for thousands of years, such as the author of <a href="http://www.nature.com/climate/2008/0812/full/climate.2008.122.html">this article</a>, is wrong. If we stopped burning fossil fuels tomorrow, the CO2 concentration of the atmosphere would return to its pre-industrial value within fifty years.<br /><br />When carbon is absorbed or emitted by the ocean, it does so as a molecule of CO2. Statistical mechanics <a href="http://homeclimateanalysis.blogspot.com/2016/01/carbon-cycle-effect-of-temperature.html">dictates that</a> the rate of absorption is weakly dependent upon temperature, but the rate of emission is strongly dependent upon temperature. When we calculate the effect of temperature upon the equilibrium between the ocean and the atmosphere, we conclude that a 1°C warming of the oceans will cause a 10 ppmv increase in the concentration of CO2 in the atmosphere. When we <a href="http://homeclimateanalysis.blogspot.com/2016/01/carbon-cycle-correlation-between.html">look back</a> at the record of CO2 concentration and temperature over the past 400,000 years, we see the correlation we expect, with the magnitude of the changes in good agreement with our prediction. For a 12°C increase in temperature, for example, the concentration of CO2 increases by 110 ppmv.<br /><br />If we consider the atmosphere of the Earth in pre-industrial times, its atmospheric CO2 concentration was roughly 300 ppmv. A more exact value for the creation of carbon-14 is 7.5 kg/yr and we conclude that 37 Pg/yr or carbon was being absorbed and emitted by the ocean. When we add 10 Pg/yr human emissions from burning fossil fuels, we expect the concentration of CO2 in the atmosphere to rise by 27% to 380 ppmv, which is close to the 400 ppmv we observe.<br /><br />Our analysis of the carbon cycle makes three independent and unambiguous predictions all of which turn out to be correct to within ±10%. Our analysis is reliable, and it tells us that it will take roughly eight thousand years to double the CO2 concentration of the atmosphere if we continue burning fossil fuels at our current rate. Assertion (2) above is wrong by two orders of magnitude. The theory of Anthropogenic Global Warming, as stated above, is untrue.<br /><br />POST SCRIPT: Assertion (1) is harder to falsify, and we do not claim to have done so in a manner convincing to all readers. Nevertheless, we did conclude that assertion (1) had to be wrong in our series of posts on the greenhouse effect, which we summarize in <a href="http://homeclimateanalysis.blogspot.com/2012/03/anthropogenic-global-warming.html">Anthropogenic Global Warming</a>. We calculated that doubling the CO2 concentration of the atmosphere will cause the Earth to warm up by 1.5°C, provided we ignore changes in water vapor and cloud cover. As the world warms up, however, water evaporates more quickly from the oceans, and we get more clouds. Clouds reflect sunlight. The warming effect of doubling CO2 concentration is reduced by an increase in cloud cover. Clouds stabilize the Earth's temperature because they become more frequent as the Earth warms up, and less frequent as it cools down. Our simulation of the atmosphere with clouds suggests that the actual warming caused by a doubling of CO2 will be 0.9°C. So far as we can tell, the climate models used by the majority of climate scientists do not account for the increase in cloud cover that occurs as the world warms up. But they do account for the increase in water vapor in the atmosphere. Clouds cool the world, but water vapor is another greenhouse gas, and warms the world. By including water vapor but excluding the increasing cloud cover, these climate models conclude that the effect of doubling CO2 concentration will be 2°C or larger.<br /><br />POST POST SCRIPT: Some readers suggest that the atmosphere-ocean system cannot be modeled with linear diffusion because the dissolved CO2 does not increase in proportion to atmospheric CO2 concentration. We address and reject their claim in an update to <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-14-probability-of-exchange.html">Probability of Exchange</a>.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com170tag:blogger.com,1999:blog-1639738090545138933.post-68033315475794504722016-01-14T22:40:00.002-05:002016-02-07T12:06:29.426-05:00Carbon Cycle: The Correlation Between Temperature and CO2In our <a href="">previous post</a> we deduced from first principles a relationship between temperature and CO2 concentration in the atmosphere of our <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-equations-and-diagram.html">carbon cycle</a>. In the appendices of this post, we show that our calculation is consistent with the <a href="https://en.wikipedia.org/wiki/Henry%27s_law#Temperature_dependence">van t'Hoff equation</a> used by chemists to predict the concentration of CO2 above a water reservoir, and to <a href="http://www.engineeringtoolbox.com/gases-solubility-water-d_1148.html">measurements</a> of the solubility of CO2 in water. In the main body of this post, we will see how our calculation compares to measurements of temperature and CO2 concentration in the Earth's atmosphere over the past half-million years.<br /><br />In our carbon cycle <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-equations-and-diagram.html">equations</a>, <i>M<sub>R</sub></i> is the mass of carbon in the oceanic reservoir, <i>M<sub>A</sub></i> is the mass of carbon in the atmosphere, <i>k<sub>R</sub></i> is the fraction of oceanic reservoir CO2 molecules that will be emitted into the atmosphere each year, and <i>k<sub>A</sub></i> is the fraction of atmospheric CO2 molecules that will be absorbed by the oceanic reservoir each year. The rate at which carbon is emitted by the oceanic reservoir is <i>k<sub>R</sub>M<sub>R</sub></i>, and the rate at which it is absorbed by the oceanic reservoir is <i>k<sub>A</sub>M<sub>A</sub></i>. At equilibrium, these two rates will be the same, so we have:<br /><br /><i>k<sub>R</sub></i><i>M<sub>R</sub></i> = <i>k<sub>A</sub>M<sub>A</sub></i> ⇒ <i>M<sub>A</sub></i> = <i>k<sub>R</sub>M<sub>R</sub></i>/<i>k<sub>A</sub></i> (Eq. 1)<br /><br />In our <a href="http://homeclimateanalysis.blogspot.com/2016/01/carbon-cycle-effect-of-temperature.html">previous post</a> we concluded that <i>k<sub>A</sub></i> is constant with temperature, while <i>k<sub>R</sub></i> increases with temperature as e<sup>−2300/<i>T</i></sup>. Our consideration of the Earth's <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">carbon-14 inventory</a> showed that <i>M<sub>R</sub></i> = 77,000 Pg. In our natural, equilibrium carbon cycle, we <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">found</a> that <i>M<sub>A</sub></i> = 650 Pg. Given that the carbon must reside somewhere, even if <i>M<sub>A</sub></i> doubles or halves with temperature, <i>M<sub>R</sub></i> will change by less than 1%. So we can assume, to the first approximation, that <i>M<sub>R</sub></i> is constant with temperature. It is <i>M<sub>A</sub></i> that varies with temperature, and it does so in proportion to <i>k<sub>R</sub></i>. <br /><br /><i>M<sub>A</sub></i> = <i>k<sub>R</sub>M<sub>R</sub></i>/<i>k<sub>A</sub></i> ∝ e<sup>−2300/<i>T</i></sup> (Eq.2)<br /><br />Aa two temperatures <i>T</i><sub>1</sub> and <i>T</i><sub>2</sub>, the equilibrium values of <i>M<sub>A</sub></i>, which we denote <i>M<sub>A</sub></i>(<i>T</i><sub>1</sub>) and <i>M<sub>A</sub></i>(<i>T</i><sub>2</sub>), are related by:<br /><br /><i>M<sub>A</sub></i>(<i>T</i><sub>2</sub>) / <i>M<sub>A</sub></i>(<i>T</i><sub>1</sub>) = <big>e<sup>−2300/<i>T</i><sub>2</sub></sup> / e<sup>−2300/<i>T</i><sub>1</sub></sup></big> = <big>e<sup>2300(1/<i>T</i><sub>1</sub>−1/<i>T</i><sub>2</sub>)</sup></big> (Eq. 3)<br /><br />Our (Eq. 3) predicts a close, positive correlation between temperature and atmospheric CO2 concentration, and it gives us an estimate of the magnitude of the change in CO2 concentration with temperature. The graph below shows atmospheric CO2 concentration provided by <a href="http://cdiac.ornl.gov/ftp/trends/co2/vostok.icecore.co2">Barnola et al.</a> and global temperature relative to today provided by <a href="http://cdiac.ornl.gov/ftp/trends/temp/vostok/vostok.1999.temp.dat ">Petit et al.</a> over the past 425,000 years as determined from the Vostok ice cores.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/Ice_Core_TC.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/Ice_Core_TC.gif" width="470"></a><br /><b>Figure:</b> Absolute Atmospheric CO2 Concentration and Relative Temperature versus Time from Vostok ice cores. Click to enlarge. Local copies of data <a href="http://www.hashemifamily.com/Kevan/Climate/Vostok_CO2.txt">here</a> and <a href="http://www.hashemifamily.com/Kevan/Climate/Vostok_T.txt">here</a>.<br /><br />We see close and sustained correlation between CO2 concentration and temperature, even as temperature varies by 12°C. If we assume today's average global temperature is 14°C = 287 K, the change from −9°C to +3°C relative to today is a swing from 278 K to 290 K. Our (Eq. 3) predicts an increase in the mass of carbon in the atmosphere by a factor of e<sup>2300(1/278−1/290)</sup> = 1.41. Because almost all carbon in the atmosphere is bound up in CO2 molecules, the concentration of CO2 in the atmosphere is proportional to the total mass of carbon in the atmosphere, so when the total mass increases by a factor of 1.41, the CO2 concentration should increase by a factor of 1.41 also. Looking at the graph, we see CO2 rising from 190 ppmv to 290 ppmv, which is a factor of 1.52. Given the many uncertainties in our calculations, and in the ice-core measurements themselves, we are well-satisfied with the agreement between our calculations and the magnitude of the CO2 concentration changes in the ice core measurements.<br /><br />We conclude that the correlation between CO2 concentration and temperature in the Earth's atmosphere over the past half million years is due to the effect of temperature upon the exchange of CO2 between the atmosphere and the oceanic reservoir of the Earth's carbon cycle. As the temperature rises, the CO2 concentration rises, and when temperature falls, the CO2 concentration falls.<br /><br /><b>Appendix 1:</b> The <a href="https://en.wikipedia.org/wiki/Henry%27s_law#Temperature_dependence">van t'Hoff equation</a> for CO2 and water states that the concentration of CO2 above the water is proportional to e<sup>−2400/<i>T</i></sup>. Our (Eq. 2) states that it is proportional to e<sup>−2300/<i>T</i></sup>. We are well-satisfied with this agreement.<br /><br /><b>Appendix 2:</b> Suppose we have pure CO2 gas at atmospheric pressure above a reservoir of water. No matter how much CO2 the water dissolves, we maintain the same pressure of CO2 above the water. In this arrangement, unlike the arrangement of our carbon cycle, the concentration of CO2 in the water can vary. Applying the same reasoning we presented in our <a href="http://homeclimateanalysis.blogspot.com/2016/01/carbon-cycle-effect-of-temperature.htm">previous post</a>, the rate at which CO2 molecules are absorbed by the water remains constant with temperature. At equilibrium, the rate at which CO2 molecules are emitted by the water must equal the rate at which they are absorbed, which means the rate of emission must also remain constant. But our calculation states that the probability of any given CO2 molecule in the water being emitted in a certain interval of time must increase with temperature as e<sup>−2300/<i>T</i></sup>. If the rate of emission is to remain constant, the concentration of CO2 in the water must decrease with temperature as e<sup>2300/<i>T</i></sup>. Only then will the rate of CO2 emission by the water, which is the product of the probability and the concentration, remain constant with temperature. We examine the plot of CO2 solubility in water versus temperature <a href="http://www.engineeringtoolbox.com/gases-solubility-water-d_1148.html">here</a>. As temperature increases from 10°C to 20°C, the solubility of CO2 in water drops from 2.5 g/kg to 1.25 g/kg, which is a factor of 0.50. Our calculation suggests that it should drop by a factor of 0.58. We are well-satisfied with this agreement.<br /><br />Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com14tag:blogger.com,1999:blog-1639738090545138933.post-79630963288068027342016-01-05T23:19:00.000-05:002016-01-14T22:41:05.593-05:00Carbon Cycle: Effect of TemperatureUp to now, we assumed that the temperature of our <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-equations-and-diagram.html">carbon cycle</a> remained constant. We never specified what the temperature of the ocean or the atmosphere was. Nor did we assume that the temperature of the ocean or the atmosphere was uniform. We merely assumed that the temperature in each part of the system remained constant. Today we estimate how the <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-rate-of-transfer.html">rate of transfer</a> coefficients of our carbon cycle, <i>k<sub>A</sub></i> and <i>k<sub>R</sub></i>, will change with temperature. We recall that <i>k<sub>A</sub></i> is the fraction of atmospheric CO2 molecules that will be absorbed by the oceanic reservoir each year, and <i>k<sub>R</sub></i> is the fraction of oceanic reservoir CO2 molecules that will be emitted into the atmosphere each year.<br /><br /><b>Claim:</b> To the first approximation, <i>k<sub>A</sub></i> is independent of temperature.<br /><br /><b>Justification:</b> Absorption of CO2 takes place at the surface of the ocean. The air pressure at sea-level is dictated by the weight of air per square meter pressing on the sea, which is roughly ten tonnes per square meter. When the atmosphere warms up, its mass does not change. The pressure at sea-level remains constant. But the air does expand. Because the air expands, the number of CO2 molecules per cubic meter decreases. Because the gas is warmer, each CO2 molecule is moving faster. Their average velocity is proportional to the square root of the absolute temperature. Suppose we increase the temperature from 14°C to 15°C. Absolute temperature increases from 287 K to 288 K. According to the <a href="https://en.wikipedia.org/wiki/Ideal_gas_law">gas law</a>, the number of CO2 molecules per cubic meter at the ocean surface decreases by 0.35%. But their velocity increases by 0.18%, which means each CO2 molecule above the ocean surface collides with the ocean 0.18% more frequently than before. Combining these two effects, there will be a net 0.18% decrease in the number of opportunities for CO2 molecules to be absorbed. To the first approximation, this is no change at all. Let us also consider whether a faster-moving CO2 molecule, having collided with the ocean, is more or less likely to be absorbed. When CO2 dissolves in water, energy is released. The CO2 molecule does not need to supply any energy in order to be dissolved. A hotter CO2 molecule has more energy, but this energy is not required by dissolution, and will not make the reaction more likely. To the first approximation, therefore, a 1°C rise in temperature will have no significant effect upon <i>k<sub>A</sub></i>.<br /><br /><b>Claim:</b> To the first approximation, <i>K<sub>R</sub></i> increases by a factor of 1.028 for each 1°C warming of the ocean surface.<br /><br /><b>Justification:</b> Emission of CO2 takes place at the surface of the ocean. When the ocean warms by 1°C, its volume increases by 0.02%, which is insignificant. As we showed above, the warmer CO2 molecules will collide with the ocean surface 0.18% more often, but this, too, is insignificant. Let us consider whether a faster-moving CO2 molecule, having reached the surface of the ocean, is more or less likely to be emitted into the atmosphere. The emission of CO2 from solution requires heat. The <a href="https://en.wikipedia.org/wiki/Enthalpy_change_of_solution">enthalpy of dissolution</a> for CO2 <a href="http://www.nist.gov/srd/upload/jpcrd427.pdf">in water</a> is −19.4 kJ/mole = −0.20 eV per CO2 molecule. Each CO2 molecule dissolving in water releases 0.20 eV of heat. Conversely, in order to be un-dissolved, a CO2 molecule needs to supply at least 0.20 eV from its own thermal energy. At 14°C, the average molecule has kinetic energy only 0.025 eV. But a tiny fraction will have, by random collisions, a much higher energy, as dictated by the <a href="https://en.wikipedia.org/wiki/Boltzmann_distribution">Boltzmann distribution</a>. In a given period of time, the number of molecules that attain energy <i>E</i> is proportional to e<sup>−<i>E/kT</i></sup>, where <i>T</i> is absolute temperature, <i>k</i> = 8.62×10<sup>−5</sup> eV/K is the <a href="https://en.wikipedia.org/wiki/Boltzmann_constant">Boltzmann constant</a>, and <i>e</i> = 2.72 is the <a href="https://en.wikipedia.org/wiki/E_(mathematical_constant)">exponential constant</a>. The rate at which any reaction requiring thermal energy <i>E</i> takes place is proportional to e<sup>−<i>E/kT</i></sup>. Thus <i>k<sub>R</sub></i> is proportional to e<sup>−<i>E/kT</i></sup>, which means there is some constant, <i>A</i>, with units of 1/yr, for which <i>k<sub>R</sub></i> = <i>A</i> e<sup>−0.02/0.0000862<i>T</i></sup> = <i>A</i>e<sup>−2300/<i>T</i></sup>. In our natural, equilibrium carbon cycle, <i>k<sub>R</sub></i> = 0.00048 1/yr. Suppose the natural, equilibrium temperature of the ocean surface is 14°C = 287 K. We can obtain the value of <i>A</i> as follows.<br /><br /><i>k<sub>R</sub></i> @ 287K = <i>A</i>e<sup>−2300/287</sup> = <i>A</i>×0.00033 = 0.00048 1/yr ⇒ <i>A</i> = 1.45 1/yr. <br /><br />We raise the temperature by 1°C to 15°C = 288 K and re-calculate the rate.<br /><br /><i>k<sub>R</sub></i> @ 288K = 1.45 e<sup>−2300/288</sup> = 0.000493 1/yr.<br /><br />The rate has increased by a factor of 0.000493/0.00048 = 1.028, which is 2.8%. If we use 10°C or 20°C as our initial ocean temperature, we get 2.9% and 2.7% rises respectively. To the first approximation, each 1°C increase in temperature increases <i>k<sub>R</sub></i> by 2.8%, regardless of our guess at the initial temperature of the ocean surface.<br /><br />We simulate the effect of a sudden 1°C warming of our carbon cycle in the following way. We begin with our natural, equilibrium atmosphere, described by our <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-equations-and-diagram.html">carbon cycle model</a>. We increase <i>k<sub>R</sub></i> by 2.8%. We set the fossil fuel emission, <i>m<sub>F</sub></i>, to zero. We obtain the following plot of atmospheric CO2 concentration versus time.<br /><br /><center><a href="http://www.hashemifamily.com/Kevan/Climate/CO2_1C_Step.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/CO2_1C_Step.gif" width=450></a><br /><b>Figure:</b> Atmospheric CO2 Concentration Following a 1°C Step Increase in Temperature.</center><br />The increase in temperature causes an immediate 2.8% increase in the rate at which CO2 is emitted by the ocean, but no increase in the rate at which it is absorbed. The additional CO2 released by the ocean builds up in the atmosphere. As the concentration of CO2 in the atmosphere increases, the rate at which it is absorbed by the ocean increases, because there are more CO2 molecules available for absorption. When the atmospheric concentration has increased by 2.8%, the rate of absorption is once again equal to the rate of emission. The carbon cycle reaches its new equilibrium in roughly one hundred years. <br /><br />In our next post, we will see how well our calculations agree with the many empirical observations of carbon dioxide in solution, and with the history of our atmosphere over the past half-million years.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-50976388897689708342015-12-23T23:49:00.000-05:002015-12-24T10:30:45.729-05:00Carbon Cycle: With Ten Petagrams per YearToday we use our model of the Earth's <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-equations-and-diagram.html">carbon cycle</a> to show us what will happen if we start adding ten petagrams of carbon to the atmosphere each year by burning fossil fuels. Ten petagrams is roughly the amount we emitted in 2015, so we are going to see how this addition would affect the concentration of CO2 in the atmosphere if it began suddenly, continued for thousands of years, and was the only phenomenon affecting change in the carbon cycle. In particular, our calculation assumes that the temperature of the atmosphere and the ocean remain constant, which may not be true if rising CO2 concentration enhances the <a href="http://homeclimateanalysis.blogspot.com/2010/01/refutation-of-greenhouse-effect.html">greenhouse effect</a>.<br /><br />We begin with the carbon cycle in its natural, equilibrium <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">state</a>. The concentration of CO2 in the atmosphere is 300 ppm, which means it contains 650 Pg of carbon. The oceanic carbon reservoir, meanwhile, contains 77,000 Pg. Each year, the ocean emits 37 Pg into the atmosphere, and absorbs 37 Pg from the atmosphere. But now we start adding an extra 10 Pg/yr to the atmosphere by burning fossil fuels. In the the <a href="http://www.hashemifamily.com/Kevan/Climate/CC_Numerical.gif">numerical equations</a> that describe our carbon cycle, we set <i>m<sub>F</sub></i> = 10 Pg/yr. You can download our carbon cycle spreadsheet <a href="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.zip">here</a>. In it, you will find the following plot, along with our calculations.<br /><br /><center><a href="http://www.hashemifamily.com/Kevan/Climate/CO2_Absorb_10Pgyr.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/CO2_Absorb_10Pgyr.gif" width=470></a><br /><b>Figure:</b> Atmospheric and Oceanic Carbon Mass versus Time, with Ten Petagrams per Year Fossil Fuel Emissions. The time scale is logarithmic. Note that 1 Eg = 1,000 Pg. Click to enlarge.</center><br />During the first ten years, the total mass of carbon entering the atmosphere each year is 47 Pg/yr. But the mass absorbed by the oceans remains 37 Pg/yr. The mass of carbon in the atmosphere increases by 10 Pg/yr. <br /><br />After ten years, we have emitted 100 Pg by burning fossil fuels. The mass of carbon in the atmosphere has increased by 80 Pg to 730 Pg. The rate at which carbon is absorbed by the ocean has increased <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-rate-of-transfer.html">in proportion</a>, because there are more CO2 molecules available to absorb. The ocean now absorbs 41 Pg/yr. With 37 Pg/yr emitted by the ocean, and 10 Pg/yr emitted by burning fossil fuels, the carbon mass of the atmosphere increases by 6 Pg/yr.<br /><br />After a hundred years, we have emitted 1,000 Pg by burning fossil fuels. The mass of carbon in the atmosphere has increased by 180 Pg to 830 Pg. The rate at which atmospheric carbon enters the oceanic reservoir is 47 Pg/yr. The mass of the oceanic reservoir itself has increased by 820 kg to 77,820 kg, which is only 1%. The rate at which oceanic carbon emerges into the atmosphere is still close to 37 Pg/yr. The atmosphere is in equilibrium: carbon enters at 47 Pg/yr and leaves at 47 Pg/yr. Its carbon mass increases only by 0.1 Pg/yr.<br /><br />After one thousand years, the mass of carbon in the oceanic reservoir has increased by 13% to 87,000 Pg. The rate at which the ocean emits carbon into the atmosphere has increased by 13% to 42 Pg/yr. The mass of carbon in the atmosphere has risen to 910 Pg, and every year 52 Pg of atmospheric carbon is absorbed by the oceanic reservoir. The atmosphere is still in equilibrium with the ocean: each year 52 Pg enters and 52 Pg leaves. Its carbon mass continues to increase by only 0.1 Pg/yr, while the carbon in the oceanic reservoir increases by 10 Pg/yr.<br /><br />After six thousand years, the mass of carbon in the atmosphere has doubled, and the mass in the oceanic reservoir has almost doubled. The plot below shows how atmospheric CO2 concentration increases with time for the same scenario.<br /><br /><center><a href="http://www.hashemifamily.com/Kevan/Climate/CO2_Concentration_10Pgyr.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/CO2_Concentration_10Pgyr.gif" width=470></a><br /><b>Figure:</b> Atmospheric CO2 Concentration in Response to Emission of 10 Pg/yr of Carbon by Burning Fossil Fuels. Units are parts per million by volume. Click to enlarge.</center><br />Burning fossil fuels at the rate we are going today, it would take 100 years to raise the concentration of CO2 in our natural, equilibrium atmosphere from 300 ppmv to 400 ppmv, and 6,000 years to raise it from 300 ppmv to 600 ppmv.<br />Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-82079091975548085932015-12-20T22:57:00.000-05:002015-12-20T22:57:45.395-05:00Carbon Cycle: Equations and DiagramSuppose human beings start to add carbon to our natural, equilibrium atmosphere at a rate <i>m<sub>F</sub></i> by burning fossil fuels. We will express <i>m<sub>F</sub></i> in petagrams of carbon per year, or Pg/yr. Note that we have been working with carbon masses, not CO2 masses, but we can convert atmospheric carbon mass to atmospheric CO2 mass simply by multiplying by 3.7, which is the ratio of the molar mass of CO2 to the atomic mass of carbon-12. With the addition of <i>m<sub>F</sub></i>, our carbon cycle now looks like the diagram below. The time <i>t</i> = 0 yr is the moment just before we start we start adding <i>m<sub>F</sub></i> to the atmosphere.<br /><br /><center><img src="http://www.hashemifamily.com/Kevan/Climate/CC_with_Human.gif"></center><br />Or, expressed as two differential equations, it looks like this:<br /><br /><center><img src="http://www.hashemifamily.com/Kevan/Climate/CC_with_Human_Equ.gif"></center><br />We can solve these differential equations in the same way we <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html">already solved</a> those of carbon-14 concentration. Examining the equations, we see that we can obtain the behavior of non-radioactive carbon by setting the decay constant, γ, to zero and inserting the the human emission of carbon in place of the cosmic ray creation of carbon-14. More convenient than the analytic solution for our purposes, however, is a numerical solution that we can implement in a spreadsheet and combine with the historical and projected values for <i>m<sub>F</sub></i> as our study progresses.<br /><br />In the numerical solution, we pick a time step small enough that changes in the masses and transfer rates are negligible during the step. Suppose our step is δ<i>t</i>. Provided δ<i>t</i> is small enough, we can assume, for example, that <i>m<sub>A</sub></i> during each step is equal to <i>k<sub>A</sub></i> times the value of <i>M<sub>A</sub></i> at the beginning of the step. We do not have to account for the fact that <i>M<sub>A</sub></i> may be changing during the step, because the time step is so small these changes will be negligible compared to <i>M<sub>A</sub></i>. For our carbon cycle, it turns out that one year is always a small enough time step, and in long, slow developments, ten years is small enough. <br /><br />The following equations are the numerical equivalents of our differential equations. They show how we calculate <i>M<sub>A</sub></i> and <i>M<sub>R</sub></i> at time <i>t</i>+δ<i>t</i> using their values at time <i>t</i>.<br /><br /><center><img src="http://www.hashemifamily.com/Kevan/Climate/CC_Numerical.gif" width=450></center><br />The current rate at which humans emit carbon by burning fossil fuels is <a href="http://co2now.org/Current-CO2/CO2-Now/global-carbon-emissions.html">close to</a>, 10 Pg/yr. In our next post, we will set <i>m<sub>F</sub></i> to a constant 10 Pg/yr and calculate how our natural, equilibrium atmosphere responds over ten thousand years.<br /><br />PS. If you find our carbon cycle drawing too confusing and drab, you can try <a href="http://www.hashemifamily.com/Kevan/Climate/CC_by_Q.jpg">this one</a> drawn by my youngest son.<br />Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-3204757308903575532015-12-17T18:34:00.000-05:002015-12-28T18:02:44.585-05:00Carbon Cycle: Rate of TransferOur study of the radioactive isotope carbon-14, which we began in <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">Carbon-14: Origins and Reservoir</a>, led us to the <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-14-probability-of-exchange.html">conclusion</a> that roughly one in every eighteen atmospheric carbon atoms are absorbed by a vast oceanic reservoir each year. When the atmosphere and the reservoir are in equilibrium, the same amount of carbon flows out of the reservoir as into it, so that the mass of carbon in the atmosphere remains constant. But the oceanic reservoir contains over one hundred times as much carbon as the atmosphere.<br /><br />We now embark upon a series of posts in which we calculate the effect of mankind's carbon emissions upon the atmospheric CO2 concentration. Our starting point for these calculations will be the natural, equilibrium atmosphere and oceans of the late nineteenth century. This atmosphere contains 300 ppmv CO2, which implies a total atmospheric carbon mass of 650 Pg (see <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">here</a>). The oceanic reservoir contains 77,000 Pg, and each year 37 Pg of carbon is exchanged between the atmosphere and the reservoir (see <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">here</a>). <br /><br />As we showed in our <a href="http://homeclimateanalysis.blogspot.com/2015/12/carbon-14-probability-of-exchange.html">previous post</a>, the probability of an atmospheric carbon atom being transferred into the oceanic reservoir is independent of the number of carbon atoms in the atmosphere. The number of carbon atoms moving into the reservoir is equal to the total number of carbon atoms in the atmosphere divided by eighteen. If we have twice as many carbon atoms, the rate at which they move into the oceanic reservoir will double. If <i>m<sub>A</sub></i> is the mass of carbon moving into the reservoir every year, and <i>M<sub>A</sub></i> is the mass of carbon in the atmosphere, we must have:<br /><br /><i>m<sub>A</sub></i> = <i>k<sub>A</sub></i><i>M<sub>A</sub></i>, where <i>k<sub>A</sub></i> = 37 Pg/yr ÷ 650 Pg = 0.057 Pg/yr/Pg.<br /><br />The probability of a carbon atom in the oceanic reservoir being released into the atmosphere is likewise independent of the number of carbon atoms in the reservoir. If <i>m<sub>R</sub></i> is the mass of carbon leaving the reservoir every year and <i>M<sub>R</sub></i> is the mass of carbon in the reservoir, we must have:<br /><br /><i>m<sub>R</sub></i> = <i>k<sub>R</sub></i><i>M<sub>R</sub></i>, where <i>k<sub>R</sub></i> = 37 Pg/yr ÷ 77,000 Pg = 0.00048 Pg/yr/Pg.<br /><br />If we were to double suddenly the mass of carbon in the atmosphere, carbon would start to move into the oceanic reservoir at double the rate. The graph of atmospheric carbon dioxide concentration would look almost exactly like the graph of carbon-14 concentration after the nuclear bomb tests, which we present <a href="http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-bomb-tests.html">here</a>. The movement of carbon between the atmosphere and the oceanic reservoir is governed by almost exactly the same equations as the movement of carbon-14. The only difference is that carbon-14 decays, while carbon-12 lasts forever.<br /><br />UPDATE: The above calculations are consistent with <a href="https://en.wikipedia.org/wiki/Henry%27s_law">Henry's Law</a> of gasses dissolving in liquids. Henry's Law applies when the concentration of gas in the liquid has reached equilibrium at a particular temperature with the concentration of gas above the liquid. At equilibrium, the rate at which the liquid emits the gas is equal to the rate at which the liquid absorbs the gas. Henry's Law states that the equilibrium concentration of the gas in the liquid at a particular temperature is proportional to the <a href="https://en.wikipedia.org/wiki/Partial_pressure">partial pressure</a> of the gas above the liquid. According to <a href="https://en.wikipedia.org/wiki/Boyle%27s_law">Boyle's Law</a>, the partial pressure of a gas at a particular temperature is proportional to the number of gas molecules per unit volume. In our rate of transfer equations, the rate at which CO2 is absorbed by the ocean is proportional to the concentration of CO2 in the atmosphere, and the rate at which CO2 is emitted is proportional to the concentration in the liquid. If we double the concentration of CO2 in the atmosphere, our equations tell us that the rate of emission by the oceans will equal the rate of absorption only if the concentration of CO2 in the oceans also doubles, which is precisely what Henry's Law requires. Of course, we have not yet considered how changing the temperature of the atmosphere and ocean will affect the rates of transfer, but we will get to that later.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-27428965731703963832015-12-09T20:42:00.000-05:002016-11-19T19:58:07.691-05:00Carbon-14: Probability of ExchangeIn the <a href="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.jpg">carbon cycle</a> of our natural, equilibrium <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">atmosphere</a>, each carbon atom in the atmosphere has a certain probability each year of being absorbed by the reservoir. We call this the <i>probability of exchange</i> in the atmosphere. According to our <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">calculations</a>, 37 Pg of carbon is absorbed by the reservoir each year. Meanwhile, the total mass of carbon in the atmosphere is 650 Pg. To the first approximation, the probability of exchange in the atmosphere is 5.7% per year. <br /><br />Likewise, a carbon atom in the reservoir has a probability of being released into the atmosphere each year. We have 37 Pg of carbon emerging from the reservoir each year, and the reservoir contains 77,000 Pg of carbon, so the probability of exchange for the reservoir is 0.048% per year.<br /><br />Almost all carbon in the atmosphere is <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">bound up</a> in CO2. In our <a href="http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-reservoir-is-ocean.html">previous post</a> we showed that the reservoir is the deep ocean. When a carbon atom enters the deep ocean, it does so as part of a CO2 molecule. The CO2 molecule arrives by chance at the ocean surface, and by further chance it dissolves into the salty water. The CO2 molecule turns into some kind of carbonate ion. This ion mixes down through the top thousand meters of water until it reaches the deep ocean. The carbon atom is now part of the reservoir. The probability of this happening each year is 5.7% for each and every CO2 molecule in the atmosphere. <br /><br />Likewise, the probability of any carbon atom in the reservoir emerging into the atmosphere as part of a new CO2 molecule each year is 0.048%. The molecule or ion containing the carbon atom mixes up through the top one thousand meters of the ocean, arrives by chance at the surface, and by further chance emerges from the surface as an atmospheric CO2 molecule. (See UPDATE below for discussion of ocean chemistry.)<br /><br />The exchange of carbon between the atmosphere and the ocean is a <a href="https://en.wikipedia.org/wiki/First-order_reaction">first-order</a> chemical process. During the process, each carbon atom is acting alone. It does not require the cooperation of any catalyst to permit it to be dissolved in saltwater or released from saltwater. If we were to double the number of CO2 molecules in the atmosphere, so that six hundred out of every million air molecules were CO2 instead of only three hundred, the probability of any one of them being absorbed by the reservoir each year would remain the same. <br /><br />If the reservoir were something more complex than the ocean, such as a forest, we would be unable to assert that the probability of exchange was unaffected by the number of CO2 molecules in the atmosphere. A forest needs water and sunlight to convert CO2 into sugar and oxygen. If we double the number of CO2 molecules in the atmosphere, we might find that CO2 molecules are lining up inside forest leaves waiting for enough water and sunlight to arrive before they are turned into plant matter. But our reservoir is the ocean, and entering and leaving it is a statistical process in which each carbon atom acts in isolation.<br /><br />So far, we have assumed that the atmosphere and reservoir are staying at the same temperature. They could be at different temperatures, but they are neither warming nor cooling. But we note that the probability of exchange is strongly affected by temperature. We have only to look at the decreasing solubility of CO2 in water with temperature, as presented <a href="http://pubs.acs.org/doi/pdf/10.1021/ja01861a033">here</a>, to see that this strong effect must exist.<br /><br />For now, we assume our natural, equilibrium atmosphere, and its carbon reservoir, are neither cooling nor warming. The probability of exchange in the atmosphere remains constant at 5.7% per year, even if we halve or double the atmosphere's CO2 concentration. The probability of exchange in the reservoir remains constant at 0.048% per year, even if the mass of carbon in the reservoir halves or doubles. The probability of exchange is independent of concentration.<br /><br />This concludes our series of posts on carbon-14. In our upcoming posts, we will apply what carbon-14 has taught us about the Earth's carbon cycle to predict how human CO2 emissions will affect the CO2 concentration of the atmosphere.<br /><br />UPDATE [08-NOV-16]: In many gas-liquid systems, changes in concentration or acidity can change the probability of emission for a dissolved gas molecule. This variation in probability is possible when the dissolved gas appears as several species in the liquid, some of which cannot emit a gas molecule, while others can. When the relative concentrations of these species changes, the probability of a dissolved gas molecule being emitted also changes. For example, if a gas dissolves into two species A and B in equal proportion, and A has a 10% per year probability of emission while B has a 0% probability, the average probability is 5% per year. If we add acid to the system and the ratio of the two becomes 75% A and 25% B, the probability of emission for each species remains the same, but the average probability rises to 7.5% per year. Our carbon-cycle model is based upon the assumption that the atmosphere-ocean system does not exhibit concentration-dependent nor acidity-dependent probability of emission. Let us justify our assumption with a brief discussion of ocean chemistry. <br /><br />The top layer of the ocean <a href="http://www.dwc.knaw.nl/DL/publications/PU00014346.pdf">is saturated</a> with calcium carbonate (CaCO3). The CaCO3 co-exists with the carbonate ions created by CO2 dissolved in water. In Figure 5.6 of <a href="http://lawr.ucdavis.edu/classes/ssc102/Section5.pdf">Carbonate Equilibria</a> we see the pH of a liquid saturated with calcium carbonate is around 8.5 (log of the H<sup>+</sup> concentration is −8.5) for CO2 partial pressure of 300 ppmv (log of CO2 partial pressure is −3.5). The pH of our contemporary ocean is around 8.2, while the pH of a system of only CO2 and water is around 5.8. Continuing with Figure 5.6, for CO2 partial pressures in the range 100 ppmv to 10,000 ppmv (log of CO2 partial pressure is −4 to −2) the carbon content of the solution is dominated by HCO3<sup>−</sup>. The concentration of HCO3<sup>−</sup> increases in proportion to the partial pressure of CO2 (its slope is 1.0 in the log-log plot). The concentration of HCO3<sup>−</sup> is 2.0 times that of Ca<sup>2+</sup> throughout the range 100 ppmv to 10,000 ppmv (see Table 5.1 Case 2 for numerical values). As we increase the partial pressure of CO2, an equal number of of CaCO3 and CO2 molecules dissolve. Each CaCO3 molecule that dissolves adds one Ca<sup>2+</sup> ion, one HCO3<sup>−</sup> ion, and one OH<sup>−</sup> ion to the solution. Each CO2 molecule that dissolves contributes one HCO3<sup>−</sup> ion and one H<sup>+</sup> ion. The OH<sup>−</sup> and H<sup>+</sup> ions combine to form H2O, leaving the other ions in solution. The HCO3<sup>−</sup> concentration, the dissolved CO2 concentration, and the dissolved CaCO3 concentration all increase in proportion to the partial pressure of CO2. As seawater changes temperature and pressure, the saturation concentration of CaCO3 changes, and CaCO3 can precipitate, as it does in the <a href="http://onlinelibrary.wiley.com/doi/10.4319/lo.1965.10.2.0220/pdf">Persian Gulf</a>, staining the water white.<br /><br />When a gas and liquid are at equilibrium, there is as much gas entering the liquid per unit time as there is leaving it. Because gaseous CO2 has only one species, its probability of absorption into the ocean does not vary with its concentration. When we double the concentration of atmospheric CO2 from 300 ppmv to 600 ppmv, we double the rate at which it enters the ocean. When the ocean attains a new equilibrium with the 600-ppmv CO2 atmosphere, the rate at which CO2 is emitted by the ocean must be double the rate for 300 ppmv. At the same time, using the CaCO3-CO2-water system as our guide, we see that the concentration of dissolved CO2 will double for this doubling of atmospheric concentration. The doubling of dissolved concentration combined with the doubling of emission tells us that the probability of emission for CO2 molecules in the ocean is constant from 300 ppmv to 600 ppmv. <br /><br />Another way to model the atmosphere-ocean system is to ignore the calcium carbonate and instead use a CO2-water system with added OH<sup>−</sup>, such as might come from mixing NaOH with the water. We add OH<sup>−</sup> in order to increase the pH of the system from 5.8, which applies to the CO2-water system alone, to 8.2, which applies to the ocean. This OH-CO2-water system exhibits more complex behavior in the range 100 ppmv to 1000 ppmv than the CaCO3-CO2-water system. The dissolved CO2 concentration does not increase in proportion to the atmospheric concentration. So far as we can tell, this OH-CO2-water model is what climate scientists are using when they conclude that the ocean will not absorb our CO2 emissions in proportion to atmospheric CO2 concentration. They express its non-linear behavior with a number they call the <a href="https://en.wikipedia.org/wiki/Revelle_factor">Revelle Factor</a>. We do not understand why they prefer an OH-CO2-water model to the CaCO3-CO2-water model, nor have we seen in the climate science literature any plots like those of Figure 5.4 or Figure 5.6 for an OH-CO2-water system. The closest we have seen any promoter of the Revelle Factor come to plotting such graphs is <a href="https://andthentheresphysics.wordpress.com/2016/11/02/ocean-co2-uptake-part-">here</a>, but that author had no explanation for why they used the OH-CO2-water system instead of the CaCO3-CO2-water system.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-49210814748351672122015-11-29T20:58:00.000-05:002015-12-02T07:28:05.717-05:00Carbon-14: The Reservoir Is the OceanUp to now, we have guessed that the carbon-14 reservoir in our <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">carbon cycle</a> is the deep ocean, where the concentration of carbon-14 is 80% of the concentration in the atmosphere. We are using 1.0 ppt (parts per trillion) as the concentration in the atmosphere, and 0.8 ppt as the concentration in the deep ocean, but we <a href="http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-absolute-and-relative.html">recognize</a> that the absolute concentration in the atmosphere is hard to measure, and might be slightly higher than 1.0 ppt. But the concentration in the deep oceans is <a href="http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf">well-known</a> to be 80% of that in the atmosphere. Today we argue that the reservoir of carbon-14 in our carbon cycle must be the deep ocean. <br /><br />Our calculations so far go like this. Every year, cosmic rays <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">create</a> 7.5 kg of carbon-14 from atmospheric nitrogen. Carbon-14 decays back into nitrogen with a half-life of 5700 years, which means 0.012% of it decays into nitrogen every year. After fifty thousand years, the total mass of carbon-14 in the atmosphere will reach <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-establishing-equilibrium.html">equilibrium</a>. The total mass of carbon-14 will be 62,500 kg, because 62,500 kg multiplied by 0.012% is 7.5 kg per year, so the rate at which carbon-14 is created by cosmic rays is equal to the rate at which the Earth's reservoir of carbon-14 decays back into nitrogen.<br /><br />The natural, equilibrium atmosphere of the early twentieth century <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">contained</a> 650 kg of carbon-14. The remainder of the Earth's 62,500 kg of carbon-14 is elsewhere, in the <i>reservoir</i> of our <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">carbon cycle</a>. Almost all carbon-14 in the atmosphere exists in CO2. By whatever means a carbon-14 atom moves in and out of the atmosphere, it does so in CO2 molecules. When one carbon-14 molecule moves into the reservoir, it does so in the company of 1÷1.0 ppt = 1.0 trillion CO2 molecules. If the reservoir is the <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-reservoir-concentration.html">deep ocean</a>, where the concentration of carbon-14 is 0.8 ppt, we can further say that, whenever a carbon-14 atom re-enters the atmosphere from the ocean, it does so in the company of 1÷0.8 ppt = 1.25 trillion CO2 molecules. <br /><br />We <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">calculated</a> that the reservoir, assuming it is the deep ocean, must contain 77,000 Pg of carbon, and that every year 37 Pg of carbon must be exchanged between the reservoir and the atmosphere, all of it moving with CO2 molecules. When 37 Pg of carbon moves from the atmosphere to the reservoir, it carries with it 37.0 kg of carbon-14. When 37 Pg of carbon moves from the reservoir to the atmosphere, it carries with it 29.6 kg of carbon-14. The difference is a net 7.4 kg of carbon-14 flowing into the reservoir every year. We add to this another 0.1 kg per year, which is the decay of the carbon-14 that remains in the atmosphere, and we arrive at a total of 7.5 kg of carbon-14 being removed from the atmosphere by exchange and decay, which gives us equilibrium with the 7.5 kg per year being created by cosmic rays.<br /><br />With the above values of reservoir size and annual mass exchange, we obtained an <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html">analytic solution</a> to the carbon-14 concentration in our natural, equilibrium atmosphere. We <a href="http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-bomb-tests.html">showed</a> that this analytic solution was in near-perfect agreement with the relaxation-time of atmospheric carbon-14 concentration following the nuclear bomb tests of the 1950s and 1960s. Thus we are confident that our analytic solution is a reliable model for predicting the behavior of carbon-14, and therefore of all carbon, in the Earth's atmosphere.<br /><br />Let us consider the possibility that the reservoir of our carbon cycle resides somewhere other than the deep ocean. The table below shows other candidates for the reservoir, their carbon-14 concentrations, and the relaxation-time of carbon-14 concentration that we would obtain if this candidate were indeed the carbon-14 reservoir. We refer to <a href="http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf">Arnold et al.</a> for our normalized concentration values, in which the atmospheric concentration is taken to be 1.0 ppt.<br /><br /><center><table border><tr> <th>Candidate</th> <th>Carbon-14<br />Concentration<br />(ppt)</th> <th>Relaxation<br />Time (yr)</th> </tr><tr><td>Ocean, Below 1000 m</td><td>0.80</td><td>17</td></tr><tr><td>Ocean, Top 100 m</td><td>0.96</td><td>3.5</td></tr><tr><td>Biosphere, Land</td><td>1.00</td><td>0.0</td></tr><tr><td>Biosphere, Ocean</td><td>0.96</td><td>3.5</td></tr><tr><td>Soil, Humus</td><td>1.00</td><td>0.0</td></tr></table></center><br />The above candidate reservoirs are the only ones known to us that exchange CO2 with the atmosphere. The relaxation-time of carbon-14 concentration after the nuclear bomb tests was roughly 15 years. None of the above candidates are even close to being consistent with the aftermath of the bomb tests, except for the deep ocean, which is in excellent agreement. We conclude that the reservoir of our carbon cycle is the deep ocean and only the deep ocean. The other candidate reservoirs do exchange CO2 with the atmosphere, but whatever effect they have upon the Earth's carbon cycle is dwarfed by the flow of carbon into and out of the deep ocean.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-28877179040253536172015-11-19T23:15:00.002-05:002015-11-29T19:50:08.088-05:00Carbon-14: Absolute and Relative ConcentrationSuppose we want to measure the carbon-14 concentration in a cubic meter of gas. If we can measure the rate at which the gas emits beta particles, we will know the number of carbon-14 decays occurring per second. Because 0.012% of carbon-14 atoms undergo beta decay every year, we can deduce the number of carbon-14 atoms in the gas from the decay rate. By some chemical procedure, we measure the amount of carbon in the gas, and so we can deduce the carbon-14 concentration.<br /><br />To measure the rate at which the gas produces beta particles, we put the gas in a chamber. All around the chamber we arrange devices that detect beta particles. We will not be able to cover the entire surface area with beta detectors, but we can measure the fraction we cover. If we cover half the area, we can assume half the beta particles emerging from the gas will strike one of our detectors. The detectors may not detect every beta particle that strikes them. But we can perform experiments to measure their efficiency. We end up with a scaling factor by which we can multiply the number of beta particles we detect to obtain the number of beta particles that left the gas volume.<br /><br />The beta particles have energy <a href="https://en.wikipedia.org/wiki/Carbon-14">up to</a> 156 keV, with a <a href="https://en.wikipedia.org/wiki/Beta_decay">Fermi-Dirac</a> distribution. The most energetic of them can <a href="http://alignment.hep.brandeis.edu/Irradiation/Apparatus.html#esp">penetrate</a> 100 mm of air. But they are not certain to do so. The least energetic of them can penetrate only a few millimeters of gas. Only a small fraction of the beta particles emitted by carbon-14 decaying within our cubic meter of gas will ever make it to our detectors. But we can, with the help of the continuous slowing-down <a href="http://alignment.hep.brandeis.edu/Devices/Dosimeter/HTML/Electron_Range.gif">approximation</a>, and the Fermi-Dirac distribution, estimate the fraction of beta particles that will emerge from the gas volume. And so we can obtain a scaling factor by which to multiply the number beta particles we detect to obtain the total number of beta particles emitted by the gas.<br /><br />Having performed these calculations, we start to count beta particles. If the gas contains 300 ppmv of CO2 and 1.0 ppt of carbon-14, there will be of order 2.6×10<sup>11</sup> carbon-14 atoms in the gas. Of these, 3.1×10<sup>7</sup> will decay each year, or 0.97 per second. Our chamber is 50% covered with detectors that are 50% efficient, and only 10% of our beta particles get out of the gas volume, so we expect to detect of order one beta decay every forty seconds. <br /><br />Before we make our first measurement, we evacuate our chamber, to measure the background rate of beta particles. We find that our beta detectors are detecting one or two beta particles per second. These are due to radioactive isotopes in the chamber walls, cosmic ray showers, and electronic noise.<br /><br />At this point, we realize that measuring carbon-14 in a gas is going to be hard. So we freeze the carbon dioxide out of our gas sample and put a pellet of solid CO2 into a much smaller chamber. We still have the same problems, but they are less severe. We can reduce our background beta-particle rate to one every ten seconds, while raising our carbon-14 beta rate to one per second. We will have to admit, however, that our measurement of carbon-14 concentration will be accurate to no better than ±20%. And indeed, the measurements of atmospheric carbon-14 concentration vary from 1.0 to 1.5 ppt. <br /><br />But it is much easier to measure the <i>relative</i> concentration of carbon-14 in various samples of gas, water, or wood. We put one sample in our chamber and count beta particles, then another sample. If our first measurement is 20% too high, so will the second one, because the 20% error is a feature of the chamber, not the sample. The upshot of these practical considerations is that we can say that the deep oceans have carbon-14 concentration 80% of atmospheric, and be confident to ±2%, but we cannot say what the actual deep ocean concentration is to better than ±20%.<br /><br />So far, we have assumed that the atmospheric concentration of carbon-14 is 1.0 ppt. In fact, it may be as high as 1.5 ppt. If it is 1.5 ppt, we have 980 kg of carbon-14 in our natural, equilibrium atmosphere, instead of the 650 kg we calculated assuming 1.0 ppt. This 980 kg is still negligible compared to the 62,500 kg of carbon-14 in the reservoir. And it is the relative concentration in the deep ocean that dictates the values of the mass exchange rate and reservoir size in our carbon cycle model, not the absolute concentration.<br /><br />Thus, it may be that the atmosphere contains 1.5 ppt of carbon-14, not 1.0 ppt, but this has no significant effect upon our conclusions. We will continue to assume that our natural, equilibrium atmosphere contains 1.0 ppt, and quote concentrations in other potential reservoirs of carbon-14 as a fraction of the atmospheric concentration, just as the authors do in <a href="http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf">Arnold et a.</a>, where they use "relative specific activity corrected for fractionation".Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-13139445583358938052015-11-06T00:31:00.002-05:002015-11-06T00:32:13.283-05:00Carbon-14: The Bomb TestsBetween the years 1945 and 1962, we detonated hundreds of atomic bombs in the atmosphere. Most of these, and certainly the <a href="https://en.wikipedia.org/wiki/Nuclear_weapons_testing">largest</a>, were detonated in the five years leading up to the <a href="https://en.wikipedia.org/wiki/Partial_Nuclear_Test_Ban_Treaty">Partial Test Ban Treaty</a> in late 1963, which banned further atmospheric detonations. Atomic explosions produce carbon-14. By 1964, bomb tests had produced enough carbon-14 to double the atmospheric carbon-14 concentration. The graph below shows atmospheric carbon-14 concentration relative to its value before the bomb tests, as measured by various groups. <br /><br /><center><a href="http://www.hashemifamily.com/Kevan/Climate/Bomb_Test_Pettersson.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/Bomb_Test_Pettersson.gif" width=480></a><br /><b>Figure:</b> Atmospheric Concentration of Carbon-14 During and After the Bomb Tests. This graph is from an essay by <a href="http://wattsupwiththat.com/2013/07/01/the-bombtest-curve-and-its-implications-for-atmospheric-carbon-dioxide-residency-time/">Pettersson</a>. The author combined measurements from several stations to produce the most complete graph we could find. For alternate plots, see <a href="http://www.hashemifamily.com/Kevan/Climate/Bomb_Test_Wikipedia.gif">here</a> and <a href="http://www.hashemifamily.com/Kevan/Climate/Bomb_Test_OPERA.jpg">here</a>.</center><br />The relaxation of the atmospheric carbon-14 concentration from its peak follows an exponential decay with time constant roughly 15 years. After fifty years, the concentration is within a few percent of its value before the bomb tests. The carbon-14 did not linger in the atmosphere. It disappeared. Let's see if our model of the Earth's carbon cycle predicts this same relaxation or not.<br /><br />In our model, carbon-14 concentrations are governed by two <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_1.gif">differential equations</a>. We already <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html">solved</a> these equations for a starting-point of zero in the atmosphere and the reservoir. Using the same procedure, we can solve the equations for a starting point of 2.0 ppt in the atmosphere and 0.8 ppt in the reservoir, which is the state the atmosphere was in at the time of the Partial Test Ban Treaty, when atmospheric tests were stopped. We obtain the following solutions for atmospheric concentration, <i>C<sub>A</sub></i>, and reservoir concentration <i>C<sub>R</sub></i>.<br /><br /><i>C<sub>A</sub></i> = 1.0 + 0.987 <i>e</i><sup>−<i>t</i>/17</sup> + 0.013 <i>e</i><sup>−<i>t</i>/8200</sup><br /><i>C<sub>R</sub></i> = 0.8 + 0.0089 <i>e</i><sup>−<i>t</i>/17</sup> − 0.0089 <i>e</i><sup>−<i>t</i>/8200</sup><br /><br />We note that this as <i>t</i> → ∞, we have <i>C<sub>A</sub></i> → 1.0 ppt and <i>C<sub>R</sub></i> → 0.8 ppt, as before. Whatever carbon-14 we add to the atmosphere, it will eventually decay, and we will be left once again with the concentrations generated by cosmic ray production. The above equations produce the graph of atmospheric and reservoir concentration shown below.<br /><br /><center><a href="http://www.hashemifamily.com/Kevan/Climate/Bomb_Test_Model.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/Bomb_Test_Model.gif" width=480></a><br /><b>Figure:</b> Carbon Cycle Model's Prediction of Atmospheric and Reservoir Carbon-14 Concentration After Sudden Doubling of Atmospheric Concentration.</center><br />Our simple, two-part, uniform-reservoir model of the carbon cycle predicts an exponential relaxation of carbon-14 concentration after the bomb tests, with a time constant of 17 years. Needless to say: agreement between the model and observation is superb.<br />Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-58402494016406769572015-10-31T23:18:00.002-04:002015-10-31T23:18:41.171-04:00Carbon-14: Assessment of Our Carbon Cycle ModelCosmic rays turn roughly 7.5 kg of atmospheric nitrogen into carbon-14 every year. Carbon-14 decays at 0.00012 kg/kg/yr. There must be 62,500 kg of carbon-14 on Earth, because 62,000 kg decays at 7.5 kg/yr. At the end of the nineteenth century, the atmosphere contained 300 ppmv CO2, or 650 Pg of carbon, of which 1 ppt was carbon-14, or 650 kg. Roughly 0.1 kg/yr of this 650 kg decayed each year, but the remaining 7.4 kg/yr had to leave the atmosphere somehow, entering some kind of carbon-14 reservoir.<br /><br />This reservoir would have to contain the remaining 62,000 kg of carbon-14 we know must exist on Earth. Given that carbon-14 is chemically identical to normal carbon, the passage of one carbon-14 atom into the reservoir implies without any ambiguity that one trillion normal carbon atoms enter at the same time. Because carbon-14 is created in the atmosphere, the concentration in the reservoir could not be greater than 1.0 ppt, which means the reservoir would have to contain at least 62,000 Pg of carbon in order for it to hold 62,000 kg of carbon-14.<br /><br />These observations are all unassailable facts of nineteenth century Earth, and indeed of Earth today. All of the quantities we have listed remain the same to within 10%, with the exception of the CO2 concentration, which has risen by 30%. We still cannot escape the conclusion that there exists a carbon reservoir on Earth of at least 60,000 Pg that is in ready communication with the atmosphere.<br /><br />Having advanced the above argument in detail, we made two assumptions about the carbon cycle in order to develop a simple model of its behavior. We assumed that the carbon-14 concentration in the reservoir was uniform, and we assumed that this concentration was 0.8 ppt, which is the concentration in the deep oceans. With these assumptions we obtained two differential equations describing the carbon-14 concentration in the Earth's carbon cycle. The equations contained two unknown quantities: the mass exchange rate between the atmosphere and the reservoir, and the total size of the reservoir. In order to make the model fit our observed atmospheric concentration and our assumed reservoir concentration, we determined that the mass exchange rate had to be 37 Pg/yr and the reservoir had to contain 77,000 Pg of carbon.<br /><br />Our one-reservoir, uniform-concentration model is already fully constrained by our observations of carbon-14 and the atmosphere. No adjustment to any of its parameters is possible without compromising its accuracy. The model predicts how the carbon-14 concentrations will respond to any change we care to imagine. In our previous post, we presented <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Est_Equ.gif">this plot</a> of how the model predicts the concentrations will develop from a starting point of zero. <br /><br />But how are we to know if these predictions are correct? What we have done so far is akin to plotting two observations on a graph, drawing a straight line through both, and declaring the straight line to be the place where all future observations will lie. It is true that there is only one straight line that we can draw through two points, but it is also true that we can draw a straight line through any two points. It is only when we have three or more points lying on the same straight line that a straight-line model becomes convincing. In the case of our model, we can always pick a value of mass exchange and reservoir size to make it fit our observations. If we are to have more confidence in our model, we must test it against other, independent observations of carbon-14 concentration. Until then, our model is merely a credible hypothesis: worth thinking about, but not conclusive.<br /><br />This inconclusive position was the one <a href="http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf">Arnold et al.</a> found themselves in when they published their study of the carbon cycle in 1956. They did the same analysis we have done, and they studied two-reservoir and three-reservoir models as well. The more complex models produced almost identical predictions to the one-reservoir model, so they concluded that the one-reservoir model was good enough. But they were unable to test the one-reservoir model with an independent observation of carbon-14 behavior in the atmosphere. No such observations existed at the time.<br /><br />Now, sixty years later, we have the observations we need to test our carbon cycle model. In the 1960s, we inadvertently performed an experiment on the entire atmosphere, in which we doubled the mass of carbon-14 it contained. Our model makes a clear and unambiguous prediction of what will happen after such an event. If this prediction is inaccurate, our model must wrong. But if the prediction is accurate, the model must be reliable, because the chance of the model being accurate by chance is close to zero.<br />Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-20530719006460214122015-10-27T20:00:00.000-04:002015-10-28T19:10:20.156-04:00Carbon-14: Establishing EquilibriumWe have so far assumed that one million years is sufficient time for the carbon-14 concentrations in our <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">carbon cycle</a> to reach their equilibrium values. That is to say: if we started with no carbon-14 in our carbon cycle, and cosmic rays added 7.5 kg/yr to the atmosphere, it would take less than one million years for the carbon-14 concentration to stabilize at 1.0 ppt in the atmosphere and 0.8 ppt in the reservoir. In our <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html">previous post</a>, we obtained the following equations for atmospheric carbon-14 concentration, <i>C<sub>A</sub></i>, and reservoir concentration, <i>C<sub>R</sub></i>, starting from <i>C<sub>A</sub></i> = <i>C<sub>R</sub></i> = 0.0 ppt at time <i>t</i> = 0 yr. The equations assume units of ppt for concentration and years for time.<br /><br /><i>C<sub>A</sub></i> = 1.0 − 0.2 <i>e</i><sup>−<i>t</i>/17</sup> − 0.8 <i>e</i><sup>−<i>t</i>/8200</sup><br /><i>C<sub>R</sub></i> = 0.8 + 0.002 <i>e</i><sup>−<i>t</i>/17</sup> − 0.802 <i>e</i><sup>−<i>t</i>/8200</sup><br /><br />The figure below shows how the two concentrations increase with time. The scale is logarithmic, which allows us to see changes in the first year as well as in the final ninety thousand years.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/C14_Est_Equ.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Est_Equ.gif" width=480></a><br /><br />After ten years, the atmospheric concentration has risen to 0.1 ppt. We have 65 kg of carbon-14 in the atmosphere, which is most of the 75 kg created by cosmic rays in ten years. The concentration in the reservoir remains close to zero (0.0002 ppt). <br /><br />After one hundred years, the atmospheric concentration has settled upon a value of 0.2 ppt. The concentration in the reservoir remains close to zero (0.009 ppt). Cosmic rays have made 750 kg of carbon-14, but only 130 kg of this remains in the atmosphere. The exchange of 37 Pg/yr of carbon between the atmosphere and the reservoir is carrying 37 Pg/yr × 0.2 ppt = 7.4 kg/yr of carbon-14 from the atmosphere into the reservoir. This state of affairs continues through the first millennium: carbon-14 created by cosmic rays flows directly into the reservoir, with no significant amount of carbon-14 being carried back out again, because the concentration in the reservoir is still close to zero. <br /><br />After a thousand years, the concentration in the reservoir has risen to 0.1 ppt. The reservoir is beginning to fill up. The concentration in the atmosphere is now 0.3 ppt. We note that 0.3 ppt is still 0.2 ppt higher than in the reservoir. The net flow of carbon-14 into the reservoir remains 7.4 kg/yr.<br /><br />After ten thousand years, the reservoir concentration is close to 0.6 ppt. In the atmosphere it is close to 0.8 ppt. We still have a net flow of 7.4 kg/yr of carbon-14 into the reservoir each year, but now this 7.4 kg/yr is the difference between 28.7 kg/yr flowing into the reservoir and 21.3 kg/yr flowing out. At the same time, the growth in the reservoir concentration begins to slow. The rate at which carbon-14 is decaying in the reservoir is starting to be significant compared to the rate at which carbon-14 is being added to the reservoir. The reservoir now contains 45,000 kg of carbon-14, and it decays at 5.4 kg/yr.<br /><br />After fifty thousand years, the concentrations have stabilized at 0.8 ppt in the reservoir and 1.0 ppt in the atmosphere. The difference is what is required to transport almost all carbon-14 created by cosmic rays into the reservoir. The reservoir now contains so much carbon-14 that the radioactive isotope decays into nitrogen at the same rate that carbon-14 is added from the atmosphere. <br /><br />The time it takes to arrive at equilibrium is far shorter than one million years. Our model indicates that the concentration of carbon-14 in the reservoir will be set by the average carbon-14 creation rate over the past fifty thousand years. The concentration in the atmosphere, on the other hand, is the sum of the reservoir concentration and an increment that is proportional to the creation rate that has prevailed for the past fifty years. If cosmic rays were to stop suddenly, the carbon-14 concentration would drop to the reservoir concentration of 0.8 ppt within fifty years, and stay at 0.8 ppt for a thousand years before the drop in reservoir concentration became significant. Conversely, if cosmic rays were to double the rate of carbon-14 creation, the concentration in the atmosphere would rise to 1.2 ppt within fifty years, and stay at 1.2 ppt for almost a thousand years before the rise in reservoir concentration became significant.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-76780870223976617622015-10-24T23:43:00.000-04:002015-10-25T19:41:48.214-04:00Carbon-14: Analytic Solution to Concentration EquationsWe can describe the origin and fate of carbon-14 with a <a href="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.jpg">diagram</a> or a pair of <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_1.gif">differential equations</a>. One of the assumptions upon which we have based our reasoning so far is that one million years is more than enough for carbon-14 concentrations to reach equilibrium. Today we solve the two differential equations with a starting point of 0.0 ppt carbon-14 in the reservoir and the atmosphere. Our solution will tell us how long it takes for equilibrium concentrations to be established. For those of you who are not interested in following the derivation, we invite you to await our next post, in which we use the solution to plot graphs of carbon-14 concentration versus time.<br /><br />We begin by referring to our equations (1) and (2) as shown <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_1.gif">here</a>. We re-arrange the equations so that all terms in <i>C<sub>A</sub></i> are on the left side of (1) and all terms in <i>C<sub>R</sub></i> are on the left side of (2). In doing so, we treat d/d<i>t</i> as if it were just another factor, which may seem odd, but it's accurate.<br /><br /><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_2A.gif" width=480><br /><br />The only variables in these two equations are <i>C<sub>A</sub></i>, <i>C<sub>R</sub></i>, and time, <i>t</i>. All other parameters are constants that we have already <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">calculated</a>. We must eliminate <i>C<sub>R</sub></i> from the (3) so as to obtain an equation in <i>C<sub>A</sub></i> and <i>t</i> alone, which we can then solve. We eliminate <i>C<sub>R</sub></i> by multiplying (3) by the same factor that we observe on the left side of (4).<br /><br /><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_2B.gif" width=480><br /><br />We do the same thing for <i>C<sub>R</sub></i>, arriving at a differential equation in only <i>C<sub>R</sub></i> and <i>t</i>.<br /><br /><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_3.gif" width=480><br /><br />At this point we pause to check the equations by considering how they behave as time approaches infinity, as shown <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_4.gif">here</a>, and we find that they appear to behave correctly. Both equations have solutions of the same form: a constant plus two decay terms.<br /><br /><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_5.gif" width=480><br /><br />We insert values for <i>m<sub>p</sub><sup>C14</sup></i>, γ, <i>M<sub>A</sub></i>, <i>M<sub>R</sub></i>, and <i>m<sub>e</sub></i> and obtain values for all five constants in our solution. The coefficients α and β dictate how rapidly the concentration evolves with time. We have α = 0.0574 and β = 0.000122. We see that α is close to the fraction of the atmospheric carbon that is exchanged with the reservoir every year, while β is close to the decay rate of carbon-14. We have 1/α = 17 yr, which is the time constant of exchanges between the atmosphere and the reservoir, and 1/β = 8,200 yrs, which is the time constant of accumulation of carbon-14 in the reservoir. The weighting factors <i>k<sub>1</sub></i> and <i>k<sub>2</sub></i> are −0.2 and −0.8 respectively. Together, they add up to −1.0 ppt. The constant term is the equilibrium value of <i>C<sub>A</sub></i>, which comes out as 1.0 ppt, which is what we expect, because we chose the value of <i>m<sub>e</sub></i> and <i>M<sub>R</sub></i> to make sure that the equilibrium concentration would be 1.0 ppt. Our equation for <i>C<sub>A</sub></i> is as follows, where concentration is in ppt and time is in years.<br /><br /><i>C<sub>A</sub></i> = 1.0 − 0.2 <i>e</i><sup>−<i>t</i>/17</sup> − 0.8 <i>e</i><sup>−<i>t</i>/8200</sup>.<br /><br />At <i>t</i> = 0, we have <i>C<sub>A</sub></i> = 0 ppt, and when <i>t</i>→∞, <i>C<sub>A</sub></i>→1.0 ppt, as we expect. The 17-yr decay term represents the flow of carbon-14 into the reservoir. The 8200-yr term represents the accumulation of carbon-14 in the reservoir. We obtain a similar solution for <i>C<sub>R</sub></i>.<br /><br /><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_6.gif" width=480><br /><br />The coefficients α' and β' are the same as α and β. But the weighting factors are different, as is the constant term. Our equation for <i>C<sub>R</sub></i> is,<br /><br /><i>C<sub>R</sub></i> = 0.8 + 0.002 <i>e</i><sup>−<i>t</i>/17</sup> − 0.802 <i>e</i><sup>−<i>t</i>/8200</sup>.<br /><br />At <i>t</i> = 0, we have <i>C<sub>R</sub></i> = 0 ppt, and d<i>C<sub>R</sub></i>/d<i>t</i> = <i>k<sub>1</sub></i>α+<i>k<sub>2</sub></i>β = 0.0 ppt/yr, and d<sup>2</sup><i>C<sub>R</sub></i>/d<i>t</i><sup>2</sup> = <i>k<sub>1</sub></i>α<sup>2</sup>+<i>k<sub>2</sub></i>β<sup>2</sup> > 0, all of which we expect, and also when <i>t</i>→∞, we have <i>C<sub>R</sub></i>→0.8 ppt, which is one of our starting assumptions. <br /><br />We are pleased to have an analytic solution for <i>C<sub>A</sub></i> and <i>C<sub>R</sub></i>. A numerical solution to the differential equations turns out to be unstable for time steps greater than ten years. We want to plot <i>C<sub>A</sub></i> and <i>C<sub>R</sub></i> over a hundred thousand years. Ten thousand steps are cumbersome in a spreadsheet. Furthermore, the analytic solution us gives more insight into the way the parameters of the carbon cycle interact to govern its behavior.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-69650703204033159852015-10-17T19:24:00.000-04:002015-10-18T17:34:07.851-04:00Carbon 14: Reservoir ConcentrationOur study of carbon-14, which began with <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">Carbon-14: Origins and Reservoir</a>, reveals that the atmosphere is <a href="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.jpg">exchanging</a> carbon dioxide with a reservoir that contains over a hundred times as much carbon dioxide as the atmosphere itself. This exchange must be taking place, because it is the only means by which carbon-14 can be transported out of the atmosphere. The concentration of carbon-14 in the reservoir must be less than in the atmosphere, or else the carbon dioxide exchange would be unable to generate a net flow of carbon-14 into the reservoir. <br /><br />We guessed that the reservoir was the deep ocean, which does have an <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">adequate capacity</a> to hold the reservoir CO2 in solution. If the deep ocean is the reservoir, its carbon-14 concentration has <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Ocean.pdf">been measured</a> to be 0.8 ppt, which is indeed less than the atmosphere's 1.0 ppt. Assuming the concentration of the reservoir is 0.8 ppt, we <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">conclude</a> that the carbon dioxide exchange rate in our natural, <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">equilibrium atmosphere</a> is 140 Pg/yr of CO2 (or 37 Pg/yr of carbon), while the reservoir holds 280,000 Pg of CO2 (or 77,000 Pg of carbon). Because our equilibrium atmosphere itself holds 2,400 Pg of CO2 (or 650 Pg of carbon), we see that 5.6% of the CO2 in the atmosphere enters the reservoir every year and is replaced by a like amount emerging from the reservoir. Furthermore, the reservoir holds 120 times as much CO2 as the atmosphere.<br /><br />In future posts, we will find that the vast size of the reservoir, combined with the ready exchange of carbon dioxide between it and the atmosphere, imply that continuing to burn fossil fuels at the rate we are today will take thousands of years to cause a doubling in the carbon dioxide concentration of the atmosphere. Before we proceed, however, let us consider how our estimate of the size of the reservoir and the magnitude of the exchange are affected by uncertainty in the one parameter that we have guessed at: the concentration of carbon-14 in the reservoir. This concentration must be less than 1.0 ppt, and we guessed it is 0.8 ppt. But perhaps it is 0.7 ppt or 0.9 ppt. Using the equations we derived <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-size-of-carbon-reservoir.html">last time</a>, we re-calculate the carbon exchange rate and the reservoir size for these concentrations.<br /><br /><center><table border><tr> <th><i>C<sub>R</sub></i><br />(ppt)</th> <th><i>m<sub>e</sub></i><br />(Pg/yr of Carbon)</th> <th><i>M<sub>R</sub></i><br />(Pg of Carbon)</th> </tr><tr> <td>0.7</td> <td>24</td> <td>88,000</td> </tr><tr> <td>0.8</td> <td>37</td> <td>77,000</td> </tr><tr> <td>0.9</td> <td>74</td> <td>69,000</td> </tr></table><b>Table:</b> Effect of Reservoir Concentration. We have reservoir concentration of carbon-14 in ppt, <i>C<sub>R</sub></i>, carbon mass exchange rate in Pg/yr, <i>m<sub>e</sub></i>, and reservoir carbon mass in Pg, <i>M<sub>R</sub></i>. Multiply carbon masses by 44/12 to get CO2 masses.</center><br />In <a href="http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf">Arnold et al.</a>, the authors present measurements of carbon-14 concentration in various layers of the ocean, in vegetation, and in soil, relative to the carbon-14 concentration in the atmosphere. These vary from 80% to 96% of the atmospheric concentration, with the deep oceans having the lowest concentration. The concentration in our reservoir must lie somewhere in the range 0.8-1.0 ppt. If the concentration is higher than 0.8 ppt, the reservoir will be slightly smaller, but the exchange rate will be much higher. At 0.9 ppt, for example, the exchange rate will be double what it is for 0.8 ppt. We see that our estimate of the exchange rate, being based upon a reservoir concentration of 0.8 ppt, is a conservative one, while our estimate of the reservoir size is bound to be close. We can now proceed with confidence in our analysis, knowing that we are certain not to over-estimate the exchange of carbon dioxide between the atmosphere and the reservoir.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com6tag:blogger.com,1999:blog-1639738090545138933.post-60112822776746804042015-10-13T18:42:00.002-04:002015-10-18T17:18:19.886-04:00Carbon 14: Size of the Carbon ReservoirCarbon-14 is produced at roughly 7.5 kg/yr in the atmosphere. Its atmospheric concentration is one part per trillion (1.0 ppt). Its half-life is 5700 years. These three observations imply the existence of the carbon cycle we described in our <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">previous post</a>. Our <a href="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.jpg">graphical representation</a> of the carbon cycle uses the quantities listed in the table below. Together, these quantities provide a complete description of the cycle. Their numerical values apply to our natural, equilibrium atmosphere, which <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">is based upon</a> the Earth's atmosphere at the end of the nineteenth century. So far, we have values for all but two of these quantities: the mass of carbon in the reservoir, and the mass of carbon exchanged each year between the reservoir and the atmosphere. <br /><br /><center><table border><tr> <th>Symbol</th> <th>Quantity</th> <th>Value</th> </tr><tr> <td><i>m<sub>p</sub><sup>C14</sup></i></td> <td>production rate of carbon-14 in atmosphere</td> <td>7.5 kg/yr</td> </tr><tr> <td>γ</td> <td>decay rate of carbon-14</td> <td>0.00012 kg/kg/yr</td> </tr><tr> <td><i>M<sub>A</sub></i></td> <td>mass of carbon in the atmosphere</td> <td>650 Pg</td> </tr><tr> <td><i>C<sub>A</sub></i></td> <td>concentration of carbon-14 in atmosphere</td> <td>1.0 ppt at<br />equilibrium</td> </tr><tr> <td><i>M<sub>R</sub></i></td> <td>mass of carbon in the reservoir</td> <td>unknown, Pg</td> </tr><tr> <td><i>m<sub>e</sub></i></td> <td>exchange rate of CO2 between atmosphere and reservoir</td> <td>unknown, Pg/yr</td> </tr><tr> <td><i>C<sub>R</sub></i></td> <td>concentration of carbon-14 in reservoir</td> <td>0.8 ppt at<br />equilibrium</td> </tr></table><b>Table:</b> Carbon Cycle Quantities. We have 1 Pg = one petagram = 10<sup>12</sup> kg = 10<sup>15</sup> g.</center><br />Another way to represent the carbon cycle is with two differential equations, as shown below, where d/d<i>t</i> represents the rate of change of a quantity with time, where <i>t</i> is time in years. Thus d<i>C<sub>A</sub></i>/d<i>t</i> is the rate of change of the concentration of carbon-14 in the atmosphere with time, in units of ppt/yr.<br /><br /><center><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_1.gif" width=480></center><br />When our carbon cycle reaches equilibrium, the concentration of carbon-14 in the atmosphere and the reservoir settles to a constant value, so the d/d<i>t</i> terms are zero. This observation reduces our two differential equations to two simple algebraic equations, and we can solve for our two unknown quantities.<br /><br /><center><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Derivation_1A.gif" width=480></center><br />We have 37 Pg of carbon transported into the reservoir every year, including 37.0 kg of carbon-14 because the atmosphere's concentration of carbon-14 is 1.0 ppt. Another 37 Pg of carbon is emitted by the reservoir every year, including 29.6 kg of carbon-14 because the reservoir's concentration of carbon-14 is 0.8 ppt. The net flow of carbon-14 is 7.4 kg/yr into the reservoir (subtract 29.6 kg/yr from 37 kg/yr). The reservoir itself, meanwhile, contains 77,000 Pg of carbon, of which 62,000 kg is carbon-14. The reservoir's carbon-14 decays at 7.4 kg/yr (multiply 62,000 kg by 0.0012 kg/kg/yr ). As is required by the state of equilibrium, the net flow of carbon-14 into the reservoir each year is equal to the amount of carbon-14 that decays in the reservoir each year. We also have 0.1 kg of carbon-14 decaying in the atmosphere each year, so the total mass of carbon-14 that decays each year is 7.5 kg, which is equal to the amount that cosmic rays create, which is also required by the state of equilibrium.<br /><br />Although our quantities deal only with masses of carbon, we note that all exchanges of carbon between the atmosphere and the reservoir take place in the form of exchanges of CO2. In particular, all carbon-14 in the atmosphere is bound up in CO2. We assumed <a href="http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html">earlier</a> in our argument that the residence time of carbon carbon-14 in the atmosphere was much longer than the two months it takes for newly-created carbon-14 to react with oxygen to form CO2. We can now check the validity of this assumption: we have 650 Pg of carbon in the atmosphere, and 37 Pg of carbon flowing from the atmosphere into the reservoir each year. The residence time of a carbon-14 atom in the atmosphere is eighteen years (divide 650 Pg by 37 Pg/yr), which is much longer than two months, so our assumption turns out to be correct.<br /><br />We conclude that our natural, equilibrium atmosphere exchanges 37 Pg of carbon each year with a reservoir that contains 77,000 Pg of carbon. Furthermore, because all carbon in the atmosphere is in the form of CO2, and all exchanges of carbon with the reservoir are in the form of CO2, the atmosphere exchanges 140 Pg of CO2 with the reservoir every year, and the reservoir itself contains 280,000 Pg CO2 (multiply by carbon mass by the molecular weight of CO2 and divide by the atomic weight of carbon). Meanwhile, the 650 Pg of atmospheric carbon is contained in 2,400 Pg of atmospheric CO2. That is to say: the carbon reservoir is a CO2 reservoir, and it contains more than a hundred times as much CO2 as the atmosphere.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-21785506673940278392015-10-09T17:24:00.000-04:002015-10-18T17:17:33.385-04:00Carbon-14: The Carbon CycleWhen a carbon-14 atom is created in the atmosphere by a cosmic ray, it <a href="http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap11.html#14519">quickly combines</a> with oxygen to form carbon monoxide (CO). In a <a href="http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap11.html#14519">couple of months</a>, this carbon monoxide combines with more oxygen to form radioactive carbon dioxide (CO2). As we showed <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">previously</a>, almost all carbon-14 created in the atmosphere ends up in a reservoir outside the atmosphere. But how long does the average carbon-14 atom spend in the atmosphere before it leaves? Our <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-removal-from-atmosphere.html">equilibrium atmosphere</a> contains 650 kg of carbon-14, while 7.5 kg are created by cosmic rays every year. Almost one century's worth of carbon-14 production is stored in the atmosphere. It is most likely, therefore, that the averge carbon-14 atom spends several years in the atmosphere before it passes into the reservoir. Let us assume, for now, that the <a href="https://en.wikipedia.org/wiki/Residence_time">residence time</a> of carbon-14 in the atmosphere is much longer than the two months it takes for carbon-14 to be bound up into a CO2 molecule. We will come back and check the validity of this assumption later. For now, we assume that all carbon-14 in the atmosphere is bound up in radioactive CO2. <br /><br />The <a href="http://www.physicalgeography.net/fundamentals/7a.html">composition</a> of the atmosphere is such that over 99.5% of its carbon is in the form of CO2. When we observe that one in a trillion carbon atoms is carbon-14, this is equivalent to saying that one in a trillion CO2 molecules is radioactive CO2. But radioactive CO2 is chemically identical to normal CO2. The extra two neutrons in its carbon nucleus have no effect upon its interactions with other molecules. When one molecule of radioactive CO2 leaves an atmosphere that contains 1 ppt (one part per trillion) of carbon-14, it does so in the company of one trillion normal CO2 molecules. When one carbon-14 atom leaves the atmosphere, it does so with one trillion other carbon atoms. The reservoir of carbon-14 that <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">must exist</a> outside the atmosphere must also be a much larger reservoir of normal carbon. The concentration of carbon-14 in this reservoir cannot be greater than in the atmosphere, because the atmosphere is where the carbon-14 is created. We <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">already calculated</a> that the reservoir contains 62 Mg of carbon-14, so it must also contain at least 62,000 Pg (sixty-two thousand Petagrams) of normal carbon (divide the mass of carbon-14 by the maximum possible concentration of carbon-14 in the reservoir). The reservoir contains one hundred times as much carbon as our equilibrium atmosphere.<br /><br />In order for carbon-14 to leave the atmosphere, it must be carried by radioactive carbon dioxide, which in turn means that there must be a trillion times as much normal carbon dioxide leaving the atmosphere. But our equilibirum atmosphere is, by assumption, in in equilibrium. Its carbon dioxide content, and therefore its carbon content, is constant. If <i>m<sub>e</sub></i> kilograms of carbon leave the atmosphere every year and enter the reservoir, we must have the same <i>m<sub>e</sub></i> kilograms of carbon leaving the reservoir and entering the atmosphere every year. Thus the carbon content of the reservoir remains constant as well. <br /><br />If the carbon-14 concentration of the reservoir were the same as the atmosphere's, we would have the same amount of carbon-14 leaving the atmosphere as returning, because the amount of normal carbon leaving is the same as the amount returning. Therefore, the concentration of carbon-14 in the reservoir must be lower than in the atmosphere. Let the concentration in the atmosphere be <i>C<sub>A</sub></i> and in the reservoir be <i>C<sub>R</sub></i>. The net flow of carbon-14 out of the atmosphere will be <i>m<sub>e</sub></i>(<i>C<sub>A</sub></i>−<i>C<sub>R</sub></i>), which we <a href="">already calculated</a> to be 7.4 kg/yr.<br /><br />It remains for us to estimate the equilibrium concentration of carbon-14 in our carbon reservoir. It is <a href="https://en.wikipedia.org/wiki/Carbon_cycle">well known</a> that carbon is stored in vegetation and in the oceans. One kilogram of water at 14°C <a href="http://www.engineeringtoolbox.com/gases-solubility-water-d_1148.html">will hold</a> around 0.5 g of carbon in the form of dissolved CO2. Given that the mass of the oean <a href="https://en.wikipedia.org/wiki/Ocean">is roughly</a> 1.4×10<sup>21</sup> kg, the oceans have have the potential to store up to 700,000 Pg of carbon. The Earth's <a href="https://en.wikipedia.org/wiki/Biomass">biomass</a>, meanwhile, <a href="http://www.hashemifamily.com/Kevan/Climate/#Carbon%20Dioxide">appears to contain</a> only 2,000 Pg of carbon. So we will assume that the majority of the Earth's carbon reservoir is in the oceans. The concentration of carbon-14 in the deep oceans was measured by <a href="http://www.hashemifamily.com/Kevan/Climate/C14_Ocean.pdf">Bien et al.</a> to be around 80% of the concentration in the atmosphere. So we will assume that the equilibrium concentration of carbon-14 in the Earth's carbon-14 reservoir is 0.8 ppt.<br /><br />The diagram below illustrates what we have concluded so far about the exchange of carbon between the atmosphere and the reservoir. The masses of carbon in the atmosphere and the reservoir are <i>M<sub>A</sub></i> and <i>M<sub>R</sub></i> respectively. The decay rate of carbon-14 is γ.<br /><br /><center><a href="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.jpg"><img src="http://www.hashemifamily.com/Kevan/Climate/Carbon_Cycle.jpg" width=480></a></center><br />The diagram assumes that the carbon exchange between the atmosphere and the reservoir has already reached equilibrium. But it does not assume that the carbon-14 concentration has reached equilibrium. We are going to obtain an analytic solution for the evolution of carbon-14 concentration from a starting point of no carbon-14 at all. So instead of pre-supposing that we have already reached carbon-14 equilibrium, the diagram states that, given infinite time, the carbon-14 concentration in the atmosphere and reservoir will eventually reach equilibrium at 1.0 and 0.8 ppt respectively. <br /><br />What is expressed in the diagram we can write down in two differential equations with some boundary conditions. The two equations contain two unknown constants: the carbon exchange rate, <i>m<sub>e</sub></i>, and the size of the carbon reservoir, <i>M<sub>R</sub></i>. In our next post, we will use these two differential equations to deduce the values of <i>m<sub>e</sub></i> and <i>M<sub>R</sub></i>.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-17423158659548560862015-09-28T21:57:00.000-04:002015-10-18T17:17:57.396-04:00Carbon-14: Removal from the AtmosphereWe are going to estimate the mass of carbon-14 in the Earth's atmosphere prior to our burning significant quantities of fossil fuels, and prior to our detonating atomic bombs. We will use the Earth's atmosphere at the end of the nineteenth century as an approximation to the atmosphere in its natural, equilibrium state, undisturbed by human activity. <br /><br />The concentration of carbon dioxide in our equilibrium atmosphere is 300 ppmv (parts per million by volume, as estimated for the turn of the nineteenth century <a href="http://cdiac.ornl.gov/trends/co2/graphics/siple-gr.gif">here</a>). The total mass of the atmosphere is 5.3×10<sup>18</sup> kg. (Atmospheric pressure <a href="http://homeclimateanalysis.blogspot.com/2010/03/atmospheric-weight.html">is generated by</a> the weight of the atmosphere per square meter, so divide sea-level atmospheric pressure by gravitational acceleration and multiply by the surface area of the Earth to obtain atmospheric mass.) In accordance with the <a href="https://en.wikipedia.org/wiki/Ideal_gas_law">gas law</a>, the density of CO2 is 1.5 times higher than the density of air (the molar mass of CO2 is 44 g, and of air is 29 g). Thus 300 ppmv of CO2 in the atmosphere is the same as 450 ppm (parts per million by mass). The mass of CO2 in our equilibrium atmosphere is 2.4×10<sup>15</sup> kg (450 ppm of 5.3×10<sup>18</sup> kg). The molar mass of CO2 is 44 g, and that of carbon is 12 g, so the mass of carbon in the atmosphere is 6.5×10<sup>14</sup> kg (2.4×10<sup>15</sup> kg × 12 g ÷ 44 g). We will use <i>petagrams</i> (Pg) to represent large masses, where 1 Pg = 10<sup>12</sup> kg = 10<sup>15</sup> g. Our equilibrium atmosphere contains 650 Pg of carbon. <br /><br />The <a href="https://en.wikipedia.org/wiki/Carbon-14">concentration</a> of carbon-14 in our equilibrium atmosphere is 1.0 ppt (parts per trillion by mass). Almost all carbon in <a href="http://www.physicalgeography.net/fundamentals/7a.html">the atmosphere</a> is contained in CO2, so the mass of carbon-14 in our equilibrium atmosphere is 650 kg (650 Pg of CO2 × 1ppt). As we <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">already showed</a>, the equilibrium reservoir of carbon-14 on Earth is 62 Mg (7.5 kg/yr production by cosmic rays ÷ 0.00012 kg/kg/yr decay rate = 62,500 kg = 62 Mg). Of this reservoir, only 1% is to be found in the atmosphere. From now on, when we refer to the Earth's <i>carbon-14 reservoir</i> we will be referring to the 62 Mg that is <i>not</i> in the atmosphere.<br /><br />The 650 kg of carbon-14 in our equilibrium atmosphere decays at 0.078 kg/yr (650 kg × 0.00012 kg/kg/yr) and is added to by <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">cosmic ray production</a> of 7.5 kg/yr. In order for the carbon-14 content of the atmosphere to remain constant, carbon-14 must pass out of the atmosphere at 7.4 kg/yr. Let us suppose, for the sake of argument, that this 7.4 kg/yr does not pass into the carbon-14 reservoir. In that case, the 7.4 kg/yr goes somewhere else, and a new reservoir starts to build up, while the existing reservoir decays, which would mean that our carbon-14 reservoir would not be in equilibrium, which contradicts <a href="http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html">our observation</a> that the reservoir had millions of years to reach equilibrium before the nineteenth century. Thus 7.4 kg/yr of carbon-14 must pass directly from the atmosphere into the reservoir. It could be that the reservoir contains many sub-divisions communicating with one another in complex ways, but this does not alter the fact that 7.4 kg of carbon-14 is passing out of the atmosphere and into the reservoir every year.<br /><br />The figure below illustrates the origin and fate of Carbon-14 on Earth. We use <i>M</i> for mass, and <i>m</i> for mass flow. We use subscript A for atmosphere, R for reservoir, D for decay, T for transfer, and P for production. Superscript C14 means carbon-14.<br /><br /><center><img src="http://www.hashemifamily.com/Kevan/Climate/C14_Flow.gif" width=450></center><br />Where is the Earth's carbon-14 reservoir? How does it acquire 7.4 kg of carbon-14 from the atmosphere every year?Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-22082979104328059052015-09-24T12:01:00.000-04:002016-10-19T13:39:06.738-04:00Carbon-14: Origins and ReservoirThis is the first of a series of posts in which we use our knowledge of carbon-14 concentrations to arrive at firm conclusions about the way in which carbon dioxide (CO2) cycles between the atmosphere and the oceans. The implications of the atmosphere's carbon-14 concentration were studied thoroughly and objectively prior to 1960, in papers such as <a href="http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf">Arnold et al</a>. But these authors did not have available to them the results of the nuclear bomb tests of the 1960s, so their conclusions could not be as firm as ours. The same implications have been studied more recently in work such as <a href="http://euanmearns.com/whats-up-with-the-bomb-model/">Mearns</a> and <a href="http://wattsupwiththat.com/2013/07/01/the-bombtest-curve-and-its-implications-for-atmospheric-carbon-dioxide-residency-time/">Pettersson</a>, but these authors did not attend to the rate of production of carbon-14 by cosmic rays, and so did not appreciate the necessary size of the global CO2 reservoir. Modern models of the CO2 cycle are presented in papers such as <a href="http://climatemodels.uchicago.edu/geocarb/archer.2009.ann_rev_tail.pdf">Archer et al.</a>, but these models are contradicted by carbon-14 observations, so they cannot be correct.<br /><br />Carbon-14 has been produced in our atmosphere by cosmic rays for billions of years. A <a href="https://en.wikipedia.org/wiki/Cosmic_ray">cosmic ray</a> is an energetic particle arriving from space. Most are protons. Some have energy 1×10<sup>20</sup> eV. (The <a href="http://home.web.cern.ch/topics/large-hadron-collider">Large Hadronic Collider</a>, for comparison, produces protons with energy 7×10<sup>13</sup> eV.) Cosmic rays collide with atmospheric nuclei and produce showers of photons and particles. Among the particles produced are neutrons, and these neutrons can react with nitrogen-14 nuclei to produce <a href="https://en.wikipedia.org/wiki/Carbon-14">carbon-14</a>.<br /><br />A nitrogen-14 nucleus has seven protons and seven neutrons. During its reaction with a neutron, it ejects a proton but retains the neutron. The result is a nucleus with six protons and eight neutrons, which is carbon-14. The carbon-14 nucleus is unstable. Eventually, one of its neutrons will emit an electron and turn into a proton. The nucleus is once again nitrogen-14. The electron shoots out of the nucleus with energy up to 156 keV. It is called a <a href="https://en.wikipedia.org/wiki/Beta_particle">beta particle</a>, and the decay of carbon-14 is called a <i>beta decay</i>. The decay happens at random, but the probability that any given carbon-14 nucleus will decay each year is 0.012%. If we have one kilogram of carbon-14, there will be only half a kilogram left after 5700 years.<br /><br />The electrons emitted by carbon-14 decay have sufficient energy <a href="http://alignment.hep.brandeis.edu/Irradiation/Apparatus.html#esp">to penetrate</a> 50 mm of air. With care, we can measure the concentration of carbon-14 in a sample of air, or in a sample of wood, cloth, or animal tissue, by counting the electrons it produces, and weighing its carbon content. We find that one in a trillion carbon atoms in the atmosphere is a carbon-14 atom. The rest is carbon-12, with one part in a thousand carbon-13.<br /><br />Almost all the carbon-14 in our atmosphere ends up in CO2 molecules. One in every trillion atmospheric CO2 molecules contains carbon-14. The rate at which cosmic rays produce carbon-14 is of order two atoms per square centimeter of the Earth's surface per second (see <a href="http://onlinelibrary.wiley.com/doi/10.1029/RG001i001p00035/abstract">Lingenfelter</a> for measurement 2.5 atoms/cm<sup>2</sup>/s and <a href="http://arxiv.org/pdf/1206.6974.pdf">Kovaltsov et al.</a> for 1.7 atoms/cm<sup>2</sup>/s). The creation rate varies as the Earth moves through the galaxy, and with cycles of solar activity, but to the <a href="http://www.europhysicsnews.org/articles/epn/pdf/2015/02/epn2015462p26.pdf">best of our knowledge</a>, the creation rate has been constant to within ±25% over the past ten million years.<br /><br />Because we know carbon-14's rate of decay and its rate of production, which has been stable for at least a million years, we can calculate the equilibrium quantity of carbon-14 on our planet. Cosmic rays produce 2 atoms/cm<sup>2</sup>/s of carbon-14, so they produce 7.5 kg of carbon-14 every year. (Multiply 2 by the Earth's surface area in square centimeters, the number of seconds in a year, the atomic weight of carbon-14, and divide by Avogadro's number to get the number of grams produced per year.) In the past million years, cosmic rays produced 7.5 million kilograms of carbon-14. But each carbon-14 nucleus has a 0.012% chance of decaying each year, so only a small fraction of this 7.5 million kilograms still exists. Suppose 75,000 kg remained. In the coming year 9.0 kg would decay (0.012% of 75,000 kg) and only 7.5 kg would be created. The Earth's reservoir of carbon-14 would be decreasing at 1.5 kg/yr. Suppose only 50,000 kg remained. In the coming year, only 6.0 kg would decay (0.012% of 50,000 kg) and 7.5 kg would be created. The Earth's reservoir would be increasing at 1.5 kg/yr. The equilibrium size for Earth's carbon-14 reservoir is 62,000 kg (7.5 kg ÷ 0.012%). At this size, the rate at which carbon-14 in the reservoir decays is equal to the rate at which new carbon-14 is added to the reservoir by cosmic rays.<br /><br />Historically, carbon-14 atoms have been produced exclusively by cosmic rays. But in the 1960s, nuclear bomb tests doubled the concentration of carbon-14 in the atmosphere. Since then, the concentration has relaxed to its historical value. For ethical and practical reasons, it is hard to perform experiments upon the Earth's atmosphere and climate. But the doubling of the carbon-14 concentration by bomb tests amounts to a gigantic experiment upon the atmosphere, and this experiment turns out to be profoundly revealing when it comes to estimating the effect of human CO2 emissions upon the climate.<br />Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com16tag:blogger.com,1999:blog-1639738090545138933.post-67181119127539276802015-05-24T17:16:00.002-04:002015-05-24T17:18:47.715-04:00Scientific Method and Anthropogenic Global Warming<p>Suppose we have ten sick people. Dr. Quack persuades them to take his Patent Medicine. Five of them die and five of them recover. Dr. Quack says, "My medicine saved five lives!" But Dr. Nay says, "Nonsense, your medicine killed five people." Both claims are 100% consistent with the facts, and yet they are contradictory. Consistency with the facts is a necessary quality for a scientific theory, but it is not sufficient. If we allow consistency with observation to be sufficient proof of a theory, we are practicing <a href="http://en.wikipedia.org/wiki/Pseudoscience">pseudoscience</a>. The Food and Drug Administration (FDA) rejects both Dr. Quack and Dr. Nay's claims, saying the Patent Medicine has absolutely no effect upon the recovery of the patients until experiment has proved otherwise.</p> <p>The FDA has adopted the <a href="http://en.wikipedia.org/wiki/Null_hypothesis">null hypothesis</a>. The null hypothesis is the foundation of scientific reasoning. We determine the null hypothesis with <a href="http://en.wikipedia.org/wiki/Occam's_razor">Occam's Razor</a>, by which we cut off any unnecessary parts of our hypothesis until we arrive at the simplest possible explanation of our observations. And the simplest possible theory about the relationship between one thing and another is that there is no relationship at all. The only way to disprove the null hypothesis, according to scientific method, is with observations. Compelling arguments and sensible speculation are insufficient, nor is consensus among scientists, nor the authority of experts.</p> <p>When it comes to the climate, our initial null hypothesis is that the climate does not vary at all from one year to the next, nor does its carbon dioxide concentration. We will have to disprove this with observations before we can begin to discuss how humans might cause climate change. So let us look at what observations are available to disprove this hypothesis. We observe valleys carved by glaciers in warm climates. We find fossils of tropical plants in cold places. We pull out <a href="http://www.hashemifamily.com/Kevan/Climate/#Ice Cores">ice cores</a> and they suggest the Earth's temperature has varied dramatically in the past few hundred thousand years, along with its atmospheric carbon dioxide concentration (6-16°C and 200-300 ppm when measured with a 1000-year running average). These observations and many others disprove our null hypothesis. The climate does vary. It varies dramatically and naturally, whithout any human influence. This is our new null hypothesis. We call it the theory of <i>natural variation</i>.</p> <p>We can use the infra-red absorption spectrum of carbon dioxide to argue that doubling the carbon dioxide in the atmosphere will warm the planet (by roughly 1.5°C if we ignore clouds, and 0.9°C if we account for clouds, according to our own <a href="http://homeclimateanalysis.blogspot.com/2012/03/anthropogenic-global-warming.html">simulations</a>). Before we can hope to show that human carbon dioxide is affecting the climate, however, we must show that human carbon dioxide emissions affect the atmospheric carbon dioxide concentration, and on this subject our theory of natural variation states that human emissions have no effect. So we must disprove the theory of natural variation. Natural emission and absorption of carbon dioxide were already in <a href="http://www.hashemifamily.com/Kevan/Climate/#Carbon Dioxide">equilibrium</a> before man started burning fossil fuels. Our annual emissions are only 4% of the natural emissions (8 Pg/yr from burning fossil fules compared to 200 Pg/yr of natural emission). In the simplest chemical equilibriums, absorption is proportional to concentration, so our 4% increase in emission will, to the first approximation, cause atmospheric CO2 to increase by 4%, or 10 ppm. Such an increase is so slight that it's not clear how we could distinguish it from the larger natural variations.</p> <p>Even if we could prove that the recent increase from 330-400 ppm atmospheric carbon dioxide was due to our burning of fossil fuels, we would still have to prove that increasing carbon dioxide concentration causes the world to warm up. We may have a compelling reason to suspect that is has this effect, but we cannot abandon the theory of natural variation until it has been disproved, and we need observations that contradict the theory to disprove it. We cannot disprove the theory with compelling arguments alone. Even if we accept that increasing carbon dioxide traps heat somewhere in the atmosphere, this does not neccessarily mean that the climate will, as a whole, warm up. It may seem obvious that lighting a fire in my fireplace will warm my house, but the reality is that my children's bedrooms get cold when I light a fire. That's because the thermostat is near the fireplace, so when I light the fire the radiators turn off. The climate is a complex system. It could be that it contains similar surprises, where we observe the opposite effect to the one we expected. In fact, looking at the <a href="http://www.hashemifamily.com/Kevan/Climate/#Ice Cores">ice cores</a>, it appears that atmospheric carbon dioxide increases occurr one thousand years <i>after</i> increases in temperature, which could mean that carbon dioxide somehow stops the world from getting any warmer at the end of an ice age, by a process that we don't understand.</p> <p>So far as we can tell, scientific method, when applied to our observations of the Earth's climate, arrives at the assumption that the climate varies dramatically and naturally, and that human carbon dioxide emissions have no effect upon it. We look forward to seeing this assumption disproved by observations of nature, but so far we have been disappointed.</p>Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-65569188396590276162012-04-04T18:18:00.012-04:002012-04-18T09:18:40.179-04:00ConclusionThe <a href="http://homeclimateanalysis.blogspot.com/2012/03/anthropogenic-global-warming.html">anthropogenic global warming</a> (AGW) hypothesis presented by the majority of today's <a href="http://en.wikipedia.org/wiki/List_of_climate_scientists">climatologists</a> has two parts. First it claims that the world is getting exceptionally warm, and second it claims that human carbon dioxide (CO2) emissions are the cause of this warming. Seven years ago, we began our personal investigation of this hypothesis, and we did so by considering whether or not the world was indeed getting exceptionally warm. <br /><br />The first thing <a href="http://www.hashemifamily.com/Kevan/Climate/#Global%20Surface">we did</a> was estimate the uncertainty inherent in the measurements of global surface temperature. We concluded that natural variations in local climate introduce an error of roughly 0.14°C in the measurement of the change in temperature between any two points in time. The fact that the error is constant with the time over which we measure the change is a consequence of the particular <a href="http://homeclimateanalysis.blogspot.com/2009/12/trends-and-noise.html">characteristics</a> of local climate fluctuations.<br /><br />We downloaded the weather station data from <a href="http://www.ncdc.noaa.gov/oa/ncdc.html">NCDC</a> and calculated the global surface anomaly using a method we called <a href="http://www.hashemifamily.com/Kevan/Climate/#Integrated%20Derivatives">integrated derivatives</a>, but which others have called <a href="http://climateaudit.org/2010/08/19/the-first-difference-method/">first differences</a>. The graph we obtained was almost identical to the one obtained by <a href="http://www.cru.uea.ac.uk/">CRU</a> using their complex <a href="http://seaice.apl.washington.edu/Papers/JonesEtal99-SAT150.pdf">reference grid</a> method. It remains a mystery to us why institutions like CRU, <a href="http://data.giss.nasa.gov/gistemp/">NASA</a>, and NCDC use such a complex method when a far simpler one will do. All graphs show roughly a 0.6°C rise in global surface temperature from 1950 to 2000. This rise is significant compared to our expected resolution of 0.14°C.<br /><br />We made <a href="http://www.hashemifamily.com/Kevan/Climate/Num_Stations.gif">this plot</a> superimposing the number of weather stations and the global surface anomaly versus time. The number of weather stations drops dramatically from 1960 and 1990. Only one in four remain active at the end of this thirty-year period. During the same period, the global surface anomaly shows a 0.6°C rise. By selecting subsets of the weather stations, <a href="http://www.hashemifamily.com/Kevan/Climate/#Disappearing%20Stations">we found</a> that the apparent warming from 1950 to 1990 varied from 0.3°C to 1.0°C depending upon whether we used stations that disappeared in that period, persisted through that period, or existed shorter or longer intervals in the same century. Thus is seemed to us that some significant amount of work would have to be done to eliminate the change in the number of weather stations as a source of error in the data. But we saw no mention whatsoever of this source of error in published papers in which the global surface anomaly is presented, such as <a href="http://seaice.apl.washington.edu/Papers/JonesEtal99-SAT150.pdf">Jones et al.</a>.<br /><br />We plotted a <a href="http://www.hashemifamily.com/Kevan/Climate/Stations.gif">global map</a> of the available weather stations, color-coded to show the date they first started reporting. The map shows that almost all stations in the tropics began operating after 1930, while most of those in the temperate regions were operating by 1880. This seems to us to be another source of systematic error in our measurement of the global surface anomaly. <br /><br />Weather stations might also be affected by the appearance of buildings, tarmac, and road traffic. We found examples of weather stations in which such <a href="http://www.hashemifamily.com/Kevan/Climate/#Disappearing%20Stations">urban heating</a> caused an apparent warming of several degrees centigrade over a few decades. It seemed to us that this effect would have to be examined in depth by any paper presenting a global surface trend. But papers such as <a href="http://seaice.apl.washington.edu/Papers/JonesEtal99-SAT150.pdf">Jones et al.</a> do not address the urban heating issue directly. Instead, they claim that the effect is negligible and refer to other papers as proof. But when we looked up those other papers, we did not find any such proof.<br /><br />In order to argue that modern temperatures were exceptionally warm, climatologists produced the <a href="http://www.hashemifamily.com/Kevan/Climate/#Mann-Made%20Warming">hockey stick</a> graph, in which a collection of potential long-term measurements of global surface temperature were combined together under the assumption that they could be trusted only to the extent that they showed a temperatures increase from 1950 to 2000. Indeed, if a measurement showed a temperature decline in that period, the hockey stick method would <a href="http://amac1.blogspot.com/2010/08/tiljander-data-series-data-and-graphs.html">flip the trend over</a> and add it to the combination so that it now contributed to a rise in the same period.<br /><br />The hockey-stick graph shows no sign of the <a href="http://homeclimateanalysis.blogspot.com/2010/02/medieval-warm-period.html">Medieval Warm Period</a>, in which Greenland was inhabited by farmers, nor the Little Ice Age, when the Thames was known to freeze over, and nor should we expect it to. Given a random set of measurements, the hockey-stick combination method will almost always produce a graph that shows a sharp rise from 1950 to 2000 and a gentle descent during the thousand years before-hand. When applied to the existing measurements of temperature by tree rings, ice cores, and other such indirect methods, it is no surprise that the method produced that same shape.<br /><br />We presented our doubts about the surface temperature measurements and the hockey stick graph to believers in the AGW hypothesis. We were received with <a href="http://homeclimateanalysis.blogspot.com/2010/07/hockey-stick-defenders.html">disdain</a> and given no <a href="http://homeclimateanalysis.blogspot.com/2010/07/hockey-stick-defenders.html">satisfactory</a> answers. Furthermore, the <a href="http://homeclimateanalysis.blogspot.com/2009/11/climategate.html">Climategate</a> affair revealed several <a href="http://www.hashemifamily.com/Kevan/Climate/#Climategate">significant breaches</a> of scientific method by the climate science community. For example, in <a href="http://www.hashemifamily.com/Kevan/Climate/Climategate/Briffa_Continued.jpg">this graph</a> produced by climatologists for the World Health Organization, the authors removed the tree ring temperature data from 1960 onwards because it showed a decline in temperature, and substituted temperature station measurements in their place. They plotted the combination as a single line. When I asked a prominent climatologists what exactly had been done, <a href="http://www.hashemifamily.com/Kevan/Climate/Climategate/GS1.txt">he said</a>, "The smooth was calculated using instrumental data past 1960." He declared that a better way to handle the divergence of the tree-ring data from the station measurements would be to cut short the graph of tree-ring data at 1960, so as to hide the decline in temperatures measured by the tree rings.<br /><br />What we see here is the assumption by climatologists that the world has been warming up and that the global temperature measured by weather stations is correct. This assumption leads them to delete conflicting data on the grounds that it must be bad data. Thus it becomes impossible for them to discover that their assumption is incorrect. By this time, we were skeptical of the global surface anomaly we obtained from the station data. We were no longer certain that the data itself had not been modified by NCDC. We had little reason to trust any other measurement produced by climatologists, we were unimpressed with the hockey-stick method of combining measurements, and we were quite certain that recent temperatures were not exceptional for the past ten thousand years.<br /><br />We turned our attention to the second part of the AGW hypothesis: the one that says doubling the atmosphere's CO2 concentration will increase the surface temperature by roughly 3°C. It took us a long time to come to a conclusion on this one. The climate models upon which such predictions are based are private property of various climatologists. In any event, we do not trust models produced by a community that is willing to delete data that conflict with its assumptions. If they are willing to delete data, we must assume that they are willing to adjust their models until the models give predictions consistent with their AGW hypothesis.<br /><br />We began with some <a href="http://homeclimateanalysis.blogspot.com/2010/01/radiation-cooling.html">laboratory experiments</a> on radiation. We stated the principle of the <a href="http://homeclimateanalysis.blogspot.com/2010/01/refutation-of-greenhouse-effect.html">greenhouse effect</a>. After a great deal of searching around, we eventually obtained the absorption spectrum of various layers of the Earth's <a href="http://homeclimateanalysis.blogspot.com/2010/09/extended-atmospheric-absorption-spectra.html">atmosphere</a>. This allowed us to confirm that, if the skies remained clear, a doubling of CO2 concentration <a href="http://homeclimateanalysis.blogspot.com/2010/10/with-660-ppm-co2.html">would cause</a> the world to warm up by about 1.5°C.<br /><br />But of course the skies <a href="http://homeclimateanalysis.blogspot.com/2010/10/self-regulation-by-clouds.html">don't remain clear</a>. The formation of clouds is a strong function of surface temperature. If the world warms up, there will be more clouds. They will reflect more of the Sun's light, while at the same time, slowing down the radiation of heat into space by the Earth. To determine how these two effects would interact, we built our own climate model, which we called <a href="http://homeclimateanalysis.blogspot.com/2011/01/circulating-cells.html">Circulating Cells</a>.<br /><br />When it comes to determining the effect of increased cloud cover, the most critical parameter to decide upon is the reflection of sunlight by clouds per millimeter of water depth in the cloud. It seemed to us that there should be a large body of literature written recently upon this subject because it is so important to climate modeling. The best paper we found upon the subject was written in 1948, <i>Reflection, Absorption, and Transmission of Insolation by Stratus Cloud</i>. We found a couple of more recent papers about reflection, such as <a href="http://onlinelibrary.wiley.com/doi/10.1002/met.285/abstract">this one</a>, but they do not attempt to provide an empirical formula for the reflection of clouds with increasing cloud depth. We concluded that climatologists are not examining this issue in detail.<br /><br />In a long sequence of small steps, we built up our climate model until it implemented surface <a href="http://homeclimateanalysis.blogspot.com/2011/05/impetus-dissected.html">convection</a>, surface <a href="http://homeclimateanalysis.blogspot.com/2011/07/simulated-planet-surface.html">heat capacity</a>, <a href="http://homeclimateanalysis.blogspot.com/2011/09/simulated-clouds-part-i.html">evaporation</a>, <a href="http://homeclimateanalysis.blogspot.com/2011/10/clouds-without-rain.html">cloud formation</a>, <a href="http://homeclimateanalysis.blogspot.com/2011/12/simulated-rain.html">precipitation</a>, and <a href="http://homeclimateanalysis.blogspot.com/2012/01/radiating-clouds.html">radiation</a> by clouds. We tested every aspect of the simulation in detail, and based its operating parameters upon our own estimates and upon whatever measurements we could find in climate science journals. We did not choose our model parameters to suit any hypothesis of our own, nor could we have done, because we did not have a model capable of testing the AGW hypothesis until the final stage, and we did not change the parameters in that final stage.<br /><br />The <a href="http://www.hashemifamily.com/Kevan/Climate/CC_11.tcl">latest version</a> of our climate model shows that cloud cover <a href="http://homeclimateanalysis.blogspot.com/2012/02/thickening-clouds.html">increases rapidly</a> as the surface warms above the freezing point of water. The <a href="http://homeclimateanalysis.blogspot.com/2011/08/evaporation-rate.html">evaporation rate</a> of water from the surface increases approximately as the square of the temperature above freezing, and the only way for water to return to the surface is to <a href="http://homeclimateanalysis.blogspot.com/2011/12/evaporation-cycle.html">form a cloud</a> first. If we ignore the increased reflection of sunlight due to increasing cloud cover, and consider only the slowing-down of radiation into space by the same increase in cloud cover, <a href="http://homeclimateanalysis.blogspot.com/2012/03/anthropogenic-global-warming.html">our model shows</a> roughly 3°C of warming due to a doubling in CO2 concentration. But when we take account of the increased reflection of sunlight by the increasing cloud cover, the <a href="http://homeclimateanalysis.blogspot.com/2012/03/anthropogenic-global-warming.html">warming drops</a> to 0.9°C.<br /><br />It seems to us that the climate models used by climatologists ignore the reflection of sunlight due to clouds. They may allow for some fixed fraction of sunlight to be reflected by clouds, but they do not allow this fraction to increase with increasing surface temperature. Thus they conclude that the warming due to CO2 doubling will be 3°C. If they took account of the increased reflection, the effect would be far smaller and less dramatic: roughly 1°C.<br /><br />Doubling the CO2 concentration of the atmosphere will indeed encourage the world to warm up, but not by enough that we should worry. Right now CO2 concentration has increased from roughly 300 ppm to 400 ppm in the past century. If it gets to 600 ppm then we can say that the rise in CO2 concentration will tend to warm the Earth by 1°C. But we are unlikely to be able to check our calculations, because the natural variation in the Earth's climate is itself <a href="http://www.hashemifamily.com/Kevan/Climate/#Ice%20Cores">of order</a> ±1°C from one century to the next.<br /><br />And so we find ourselves at the end of our journey. Modern warming is not exceptional, and doubling the CO2 concentration will cause the world to warm up by roughly 1°C, not 3°C. The only part of the AGW theory we have not investigated is its assertion that human CO2 emissions are responsible for the increase in atmospheric CO2 concentration over the past century. <br /><br />My thanks to those of you who took part in the effort, both by private e-mails and in the comments. I would not have continued the effort without your participation. I hope it is clear that my use of "we" instead of "I" is in recognition of the fact that this has been a group effort. I will continue to answer comments on this site, and I will consider any suggestions of further work. To the first approximation, however: we're done.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com28tag:blogger.com,1999:blog-1639738090545138933.post-89704344890085744672012-03-15T13:00:00.008-04:002012-11-07T09:50:28.517-05:00Anthropogenic Global WarmingThe Anthropogenic Global Warming (AGW) hypothesis <a href="http://monthlyreview.org/2008/07/01/the-scientific-case-for-modern-anthropogenic-global-warming">states</a> that doubling the CO2 concentration of the Earth's atmosphere will raise the average surface temperature of the Earth by a minimum of 1.5°C, and more likely 3°C. <br /><br />In our <a href="http://homeclimateanalysis.blogspot.com/2010/09/total-escaping-power.html">investigation</a> of the absorption and emission of long-wave radiation by the Earth's atmosphere, we <a href="http://homeclimateanalysis.blogspot.com/2010/10/with-660-ppm-co2.html">calculated</a> that a sudden doubling of CO2 concentration would decrease the power the Earth radiates into space by 6.6 W/m<sup>2</sup>. We then estimated how much the Earth and its atmosphere would have to warm up in order to restore the heat radiated into space to its original value. We assumed there would be no significant change in cloud cover as a result of the warming, and we applied <a href="http://homeclimateanalysis.blogspot.com/2010/01/black-bodies.html">Stefan's Law</a> to calculate how the heat radiated by the Earth and its atmosphere would increase. We found that the required increase in surface temperature would be around 1.6°C. If there were no change in cloud cover, then the heat arriving from the Sun would remain the same, and we could expect the Earth to warm up by 1.6°C so as to once again arrive at thermal equilibrium.<br /><br />The AGW hypothesis states that in the event of the Earth warming, changes in cloud cover will be such as to amplify the warming we calculate using Stefan's Law. Here is an extract from today's entry on <a href="http://en.wikipedia.org/wiki/Global_warming">Global Warming</a> at Wikipedia.<br /><br /><i>The main positive feedback in the climate system is the water vapor feedback. The main negative feedback is radiative cooling through the Stefanâ€“Boltzmann law, which increases as the fourth power of temperature.</i><br /><br />As the Earth warms up, water <a href="http://homeclimateanalysis.blogspot.com/2011/08/evaporation-rate.html">evaporates more quickly</a> from the oceans. Almost all water that evaporates must turn into <a href="http://homeclimateanalysis.blogspot.com/2011/12/evaporation-cycle.html">clouds</a> before it returns to Earth. Water condensing directly onto grass in the morning is an exception to this rule, but the vast majority of water vapor will return only as rain or snow, and so must first take the form of a cloud. <br /><br />Clouds absorb <a href="http://homeclimateanalysis.blogspot.com/2010/02/extreme-greenhouse.html">long-wave</a> radiation emitted by the Earth's surface, so the Earth cannot radiate its heat directly into space. Instead, the clouds radiate into space and warm the Earth with <a href="http://homeclimateanalysis.blogspot.com/2011/08/back-radiation.html">back radiation</a>. Thus increasing cloud cover means less heat radiated into space for the same surface temperature. This is the <i>positive feedback</i> referred to by the AGW hypothesis. The AGW climate models predict that this positive feedback will amplify the minimum 1.5°C warming caused by CO2 to roughly 3°C. Some say 2°C and other say 4°C, but all agree that the actual warming will be greater than 1.5°C.<br /><br />We see this positive feedback in our <a href="http://homeclimateanalysis.blogspot.com/2011/01/circulating-cells.html">Circulating Cells</a> simulation, version <a href="http://www.hashemifamily.com/Kevan/Climate/CC_11.tcl">CC11</a>, which simulates the formation of clouds as well as their absorption and emission of long-wave radiation. The graph below shows a close-up of the behavior of the simulation in the neighborhood of its <a href="http://homeclimateanalysis.blogspot.com/2012/03/equilibrium-point.html">equilibrium point</a> for 350 W/m<sup>2</sup> solar power.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/AGW_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/AGW_1.gif" width=500></a><br /><br />The blue line shows how the power that penetrates to the surface of our simulated planet varies with increasing surface temperature. The orange line shows how the total power escaping from our simulated planet increases with surface temperature. These two lines cross at <b>a</b>, where temperature is 288 K and total escaping power is 290 W/m<sup>2</sup>. Our simulated atmosphere absorbs 50% of long-wave radiation, which is an adequate approximation of our atmosphere with its current concentration of CO2 (roughly 330 ppm). <br /><br />The green line is the same as the orange line, but displaced down by 6.6 W/m<sup>2</sup>, which is the amount by which we <a href="http://homeclimateanalysis.blogspot.com/2010/10/with-660-ppm-co2.html">calculated</a> the total power escaping from the Earth will decrease if we double CO2 concentration (to roughly 660 ppm). Thus the green line tells us the total escaping power at the same temperature if we were to double the CO2 concentration. Point <b>b</b> on the green line is 288 K, and the total escaping power is 283.4 W/m<sup>2</sup>. <br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/AGW_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/AGW_1.gif" width=500></a><br /><br />The purple line shows how the total escaping power will increase from <b>b</b> if we assume the cloud cover is constant and use only Stefan's Law to determine the heat radiated into space by the surface and atmosphere. The red line shows the solar power penetrating to the surface if we assume the cloud cover is constant. With constant cloud cover, the penetrating solar power does not change. <br /><br />The purple and red lines meet at <b>c</b>, which is 289.6 K, or 1.6°C above the previous equilibrium point. Thus our simulation shows us that the warming due to CO2 doubling, if we ignore changes in cloud cover, will be 1.6°C, which is consistent with our previous <a href="http://homeclimateanalysis.blogspot.com/2010/10/with-660-ppm-co2.html">calculation</a>.<br /><br />The green line, however, is the simulation's calculation of the total escaping power for increasing surface temperature. We see that the heat radiated into space does not increase as quickly as Stefan's Law would lead us to expect. And the reason for that is precisely the reason quoted by the AGW hypothesis: increasing cloud cover is slowing down the radiation of heat into space. The green line and the red line intersect at <b>d</b>, which is 290.7 K, or 2.7°C above our original equilibrium temperature. This is the new equilibrium temperature of the planet surface if we double CO2 concentration and we assume that there will be no change in the solar power penetrating to the surface while the cloud cover increases.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/AGW_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/AGW_1.gif" width=500></a><br /><br />But the solar power penetrating to the surface must decrease as cloud cover increases. Clouds <a href="http://homeclimateanalysis.blogspot.com/2010/10/clouds.html">reflect sunlight</a>. Thick clouds reflect 90% of solar power back into space. Even thin, high clouds reflect 10%. Increasing cloud cover will decrease the solar power penetrating to the surface. That is why our blue line slopes downwards. This is <a href="http://homeclimateanalysis.blogspot.com/2011/11/negative-feedback.html">negative feedback</a>, which acts against the positive feedback described by the AGW theory. The blue line shows how the solar power penetrating to the surface decreases as our cloud cover increases.<br /><br />The blue line and the green line intersect at <b>e</b>, which is the equilibrium point we arrive at after doubling the CO2 concentration and considering <i>both</i> the positive feedback of back-radiation <i>and</i> the negative feedback of solar reflection. The temperature at <b>e</b> is 288.9 K, which is 0.9°C above our original equilibrium temperature.<br /><br />Thus our simulation shows how the negative feedback generated by clouds dominates their positive feedback, and suggests that the actual warming of the Earth's surface due to a doubling of CO2 will be closer to 0.9°C than the 3°C predicted by the AGW hypothesis.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com5tag:blogger.com,1999:blog-1639738090545138933.post-15247209400141401452012-03-07T16:00:00.001-05:002012-03-07T16:00:03.717-05:00Equilibrium PointHeat arrives at the Earth in the from of sunlight. The clouds reflect some of it. Almost all the rest arrives at the Earth's surface. Of the sunlight that reaches the surface, some is reflected back into space, especially by snow, but most is absorbed. To the first approximation, therefore, sunlight affects the climate only by it warming the surface, not by warming the atmosphere directly.<br /><br />In our <a href="http://homeclimateanalysis.blogspot.com/2011/01/circulating-cells.html">Circulating Cells</a> simulation, version <a href="http://www.hashemifamily.com/Kevan/Climate/CC_11.tcl">CC11</a>, solar power is either reflected by clouds or absorbed by the surface. The only way for heat to enter the atmosphere is by contact with the surface, so it is the temperature of the surface that drives the the circulation of gases and the creation of clouds, snow, and rain. When we observe, for example, that 80% of solar power is penetrating to the surface when the surface gas temperature is 290 K, we can assume that the 80% of solar power will penetrate to the surface whenever the surface gas temperature is 290 K.<br /><br />Our program of <a href="http://homeclimateanalysis.blogspot.com/2012/02/solar-increase.html">solar increase</a> provides us with observations of cloud depth and penetrating power for a range of solar powers, and therefore for a range of surface temperatures. The following graph shows how the fraction of solar power penetrating to the surface varies with surface temperature.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/EP_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/EP_1.gif" width=500></a><br /><br />In our simulation we calculate the penetrating power for each surface block using the depth of the clouds in the column of cells above the surface block. We cannot, however, use the average cloud depth to obtain a measurement of average penetration fraction. When we do so, our measurement is always too low, because there are gaps in the cloud cover, and these gaps allow more light to pass through than if the clouds were distributed uniformly. <br /><br />In the following graph, we have taken our measurements of penetration fraction and used them to calculate the power that would penetrate to the surface for two values of solar power: 350 W/m<sup>2</sup> and 400 W/m<sup>2</sup>. Along with these penetrating powers, we also plot the total escaping power.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/EP_2.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/EP_2.gif" width=500></a><br /><br />At <a href="http://homeclimateanalysis.blogspot.com/2012/01/thermal-equillibrium.html">thermal equilibrium</a>, the total escaping power must be equal to the penetrating power. Thus the intersection of the penetrating power and escaping power graphs occurs at the equilibrium temperature. For solar power 350 W/m<sup>2</sup>, the intersection occurs at 288 K. For 400 W/m<sup>2</sup>, the intersection occurs at 292 K. These are, of course, the same equilibrium temperatures we obtain from our <a href="http://www.hashemifamily.com/Kevan/Climate/TC_1.gif">existing graph</a> of surface temperature versus solar power. <br /><br />In our next post, we will use the above graph and the calculations we presented in <a href="http://homeclimateanalysis.blogspot.com/2010/10/with-660-ppm-co2.html">With 660 PPM CO2</a> to estimate how much the Earth would warm up if we doubled the CO2 concentration of its atmosphere.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-44254813265358274682012-03-02T16:00:00.011-05:002012-03-02T16:00:06.151-05:00Conservation of HeatIn <a href="http://homeclimateanalysis.blogspot.com/2012/02/solar-increase.html">Solar Increase</a> we were concerned that our program of increasing solar power did not allow adequate time at each value of solar power for the simulation to arrive at thermal equilibrium. As we described in our <a href="http://homeclimateanalysis.blogspot.com/2012/02/thickening-clouds.html">previous post</a>, we now have measurements of the state of the simulation at each value of solar power. The following graph plots penetrating power, total escaping power, and downwelling power versus solar power. The <a href="http://homeclimateanalysis.blogspot.com/2012/01/thermal-equillibrium.html">penetrating power</a> is the power per square meter that gets to the surface through the clouds. The <a href="http://homeclimateanalysis.blogspot.com/2010/09/total-escaping-power.html">total escaping power</a> is the power per square meter that the surface and atmosphere together radiate into space. The <a href="http://homeclimateanalysis.blogspot.com/2012/01/radiating-clouds.html">downwelling power</a> is the power arriving back at the surface from the radiating clouds.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/CH_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/CH_1.gif" width=500></a><br /><br />The downwelling power is increasing because the cloud layer is <a href="http://homeclimateanalysis.blogspot.com/2012/02/thickening-clouds.html">getting thicker</a>. The clouds radiating heat towards the surface are lower down and therefore warmer. But it is the equality of the penetrating and escaping power that interests us the most today. At <a href="http://homeclimateanalysis.blogspot.com/2012/01/thermal-equillibrium.html">thermal equilibrium</a>, the total escaping power should be equal to the penetrating power. And indeed we see that this appears to be the case for all values of solar power.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0tag:blogger.com,1999:blog-1639738090545138933.post-12793872385443821522012-02-26T23:55:00.004-05:002012-02-27T00:00:37.946-05:00Thickening CloudsAs we describe in <a href="http://homeclimateanalysis.blogspot.com/2012/02/solar-increase.html">Solar Increase</a>, we warmed up our simulated planet by increasing the incoming solar power by 10 W/m<sup>2</sup> every two thousand hours of simulated time. Starting from an initial value of 100 W/m<sup>2</sup>, we increased the solar power to 1200 W/m<sup>2</sup> over the course of three weeks of our own time, which corresponds to the passage of over a million hours of simulated time. During the course of the simulation, we recorded the state of the atmospheric array every twenty hours, and these recordings constitute our measurements of the simulated atmosphere during the course of our simulated experiment. <br /><br />In our <a href="http://homeclimateanalysis.blogspot.com/2012/02/solar-increase-continued.html">pervious post</a> we observed that some properties of the atmosphere, such as penetrating power, fluctuated greatly from one measurement to the next. In order to reduce the influence of these fluctuations, we took the average of the last 500 hours of measurements at each value of incoming solar power, and so obtained a value for each property at each solar power. The graph below shows how some of these properties vary with solar power.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/TC_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/TC_1.gif" width=500></a><br /><br />Surface temperature increases hardly at all from 800 W/m<sup>2</sup> to 1200 W/m<sup>2</sup>, and yet cloud cover increases steadily. How can it be that cloud cover increases when the surface temperature, which drives evaporation, hardly increases at all?<br /><br />In our simulated <a href="http://homeclimateanalysis.blogspot.com/2011/12/evaporation-cycle.html">evaporation cycle</a>, precipitation beings with the formation of snow in air below temperature <i>Tf_droplets</i>. We have this parameter set to 268 K, which is five degrees below the freezing point of water. When solar power reaches 800 W/m<sup>2</sup>, the average temperature of the tropopause has reached 268 K. Snow can form only in the colder clouds of the tropopause, and nowhere below the tropopause. Each time we increase the solar power, the surface temperature at first warms a little, but within a few hundred hours, this warming reaches the tropopause, where it further slows snow formation, and increases the cloud depth. With more sunlight being reflected back into space, the surface cools again until it is hardly warmer than it started. For solar powers greater than 800 W/m<sup>2</sup>, an increase of 100 W/m<sup>2</sup> causes a substantial increase in cloud depth (roughly 0.5 mm), a slight increase in tropopause temperature (roughly 1 K), and an increase in surface temperature too small for us to detect (less than 0.3 K). <br /><br />This profound suppression of warming by our simulation is not, however, a good representation of what would happen in the Earth's atmosphere. In our simulation, gas cells that contain clouds cannot rise above our top row of cells, so there is a limit to how much they can cool down. In the Earth's atmosphere, clouds can rise as far as they need to in order to cool down and produce snow rapidly. Thus our simulation is no longer realistic once its tropopause approaches the melting point of ice. We will therefore concentrate our attention upon the behavior of the simulation for solar powers less than 600 W/m<sup>2</sup>, for which our simulated tropopause is well below the temperature required for the rapid formation of snow.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com1tag:blogger.com,1999:blog-1639738090545138933.post-4305302042344559612012-02-23T12:25:00.000-05:002012-02-23T12:25:01.999-05:00Solar Increase ContinuedToday we continue our <a href="">previous post</a> without any preamble. The graph below shows how our simulated atmosphere warms up as we increase the solar power from one hundred to twelve hundred Watts per square meter. The green line shows how we increased the solar power over the course of twelve simulated years. The blue line shows how the power penetrating to the surface varied with time. The red line is the average temperature of the air resting upon the surface of our simulated planet.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/SIC_1.gif"><img src="http://www.hashemifamily.com/Kevan/Climate/SIC_1.gif" width=500></a><br /><br />At first, when the sky is clear, the solar power and the penetrating power are equal. But when the solar power reaches 300 W/m<sup>2</sup> clouds form and the penetrating power drops below the solar power. As solar power increases from 600 W/m<sup>2</sup> to 1200 W/m<sup>2</sup>, fluctuations in the penetrating power double in their extent, but the average penetrating power appears to remain unchanged. The <a href="http://homeclimateanalysis.blogspot.com/2011/11/negative-feedback.html">negative feedback</a> generated by the <a href="http://homeclimateanalysis.blogspot.com/2011/12/evaporation-cycle.html">evaporation cycle</a> is so powerful that the surface air temperature increases by only a few degrees while we double the solar power.<br /><br />The following screen shot shows the state of the simulation after two thousand hours at 1210 W/m<sup>2</sup>. You can download this state as a text file <a href="http://www.hashemifamily.com/Kevan/Climate/SIC_1210W.txt">SIC_1210W</a> and load it into <a href="http://www.hashemifamily.com/Kevan/Climate/CC_11.tcl">CC11</a> to watch the vigorous formation of clouds and descent of precipitation.<br /><br /><a href="http://www.hashemifamily.com/Kevan/Climate/SIC_1.png"><img src="http://www.hashemifamily.com/Kevan/Climate/SIC_1.png" width=500></a><br /><br />We now have the data we need to plot graphs of surface temperature and other properties of the simulation versus solar power.Kevan Hashemihttp://www.blogger.com/profile/11014582378376549743noreply@blogger.com0