May 7, 2018

The Enhancement of Greenhouse Warming by Positive Feedback due to the Lower Solubility of CO2 Is Negligible

How much additional CO2 is released from the ocean in a positive feedback, when humans add CO2 to the atmosphere thereby increasing temperature via stronger greenhouse effect and so reducing CO2 solubility in the ocean?

A feedback release of CO2 from the oceans, enhancing the human-produced CO2 and the greenhouse warming, was sometimes mentioned as an exacerbating factor [1]. So that, if humans release CO2 (characterized by increased CO2 partial pressure of Dp1) and cause a greenhouse temperature increase DT1, the question is how much this will be exacerbated by more CO2 (Dp2) eventually released from the ocean (due to lower CO2 solubility at higher temperatures according to Henry’s law), eventually leading to additional greenhouse warming of DT2, which can eventually lead to still more CO2 release, something like a chain reaction:
If suddenly adding a CO2 increment to the atmosphere, it is supposed to result in an increased greenhouse warming, the effect being estimated, for each doubling of the CO2 concentration in the atmosphere, at between 0.25 and 5°C, usually between 0.3 and 3.0°C. The warming will push some more CO2 out of the oceans – positive solubility feedback. On the other hand, a part of said CO2 increment will dissolve in the oceans, even though dissolution of excessive CO2 in the upper surface of the oceans will take some time, and its transfer to the deeper water layers still more. Moreover, a part of the CO2 increment will be consumed by plants and by chemical reactions on the land and in the sea, and conversely, various CO2 sources like undersea volcanos will add more CO2 to the ocean and atmosphere. Assessments for real atmospheric situations are difficult, because of many involved parameters and many parallel processes, wherein the system is never in equilibrium [2]. While trying to assess the maximal possible effect of said positive feedback, regardless the dissolution of some CO2 and regardless the complicating effects, let us consider only the warming effects in order to at least roughly characterize the maximal possible effect of said positive feedback and to examine the character of the sequence DT1, DT2, DT3, etc.

The solubility of gases in water behaves according to the Henry’s law, which can be written:
c = H * p          (I)
or     p = K * c          (II)
where c is the equilibrium concentration of the dissolved gas in water [in mol/liter], p is the partial pressure of the gas above the liquid [in atm], and H and K ( K = 1/H) are called Henry constants, which depend on temperature. The relations say that the amount of dissolved gas is proportional to its pressure above the liquid, and vice versa. For CO2 and water, the law reads (for 25°C): c = 0.034 * p     and     p = 29 * c .

Let us estimate the CO2 release from the oceans (as partial pressure increase Dp) due to the increased K (or decreased H) when the temperature increases (for any reason) by DT, while assuming that c does not change much. Relation (II) may provide:
Dp/DT  =  DK/DT  *  c          (III)

1. In a first attitude, let us utilize known solubility of CO2 in water at two different temperatures for assessing DK/DT. For example from [3] or [4] we can obtain a solubility of 0.0387 mol/l at 20°C and 0.0290 mol/l at 30°C for CO2 partial pressure of 1 atm. K for 20°C and 30°C can be obtained by substituting the values into the equation (II): 1 = K20 * 0.0387 and 1 = K30 * 0.0290, yielding 25.8 and 34.5 for K20 and K30, respectively, from which DK/DT = (34.5 - 25.8)/10 = 8.7/10 = 0.87 (for temperatures around 25°C). When taking the known mean concentration of dissolved CO2 in the oceans of about 10 μmol/l [5] for c (assuming it does not change much), we can substitute into (III):
Dp/DT  =  0.87  * 10* 10-6 = 8.7 * 10-6 atm/°C = 8.7 ppm per 1°C
as 1 atm corresponds to 1 million ppm, and 10-6 atm to 1 ppm. So that a temperature increase of 1°C will raise the atmospheric CO2 content by about 8.7 ppm, if all other effects are neglected, including the increased dissolution of CO2 in the ocean.

2. In a second attitude, let us utilize the published value of DK/DT from [4], which shows DK/DT in MPa/mol fraction, at 25°C, of about 4, yielding DK/DT in atm/(mol/l) of about 0.73. Substituting to (III):
Dp/DT  =  0.73  * 10* 10-6  = 7.3 ppm ppm per 1°C.

3. In a third attitude, let us utilize the published value of dlnH/d(1/T), denoted as e, which does not change much with T and for CO2 it equals 2400 in °K [6]. By differentiating (I) we get:
dc/dT  =  p * dH/dT  +  H * dp/dT          (IV)
If c does not change much and dc/dT = 0 :
dp/dT  =  - p/H * (dH/dT)          (V)
As dlnH = dH/H, and d(1/T)/dT = -1/T2 , we can write dH/H = -e dT/T2 , from which
H/dT =  -e H/T2          (VI)
By combining (V) and (VI) we get:  dp/dT  =   (p/H) * (e H/T2)  = p * e / T2  and:
Dp/p =  e /T2 * DT          (VII)
For e = 2400 and T = 298°K:
Dp/p = 0.027 per 1°C          (VIII)
The last equation says that the relative increase of CO2 partial pressure in the atmosphere due to the decreased value of H with temperature is 0.027 = 2.7% for each deg. So, at present level of 400 ppm of CO2, the p increase per deg is (400 ppm * 0.027 =10.8):
Dp/DT = 10.8 ppm per 1°C .

4. In a fourth attitude, let us utilize the temperature changes and CO2 changes reconstructed for the last 400,000 years from the Antarctic ice cores [7]. The CO2 changes followed the temperature changes, so that it may be assumed that the CO2 release resulted from the lowered solubility in the oceans. I extracted from the below graph, manually and roughly, the CO2 increases together with the temperature increases for 16 prominent peaks. 

For example, the third peak from left provided a CO2 increase of from 200 to 300 ppm and a temperature increase of from -8.2°C to +3.4°C, yielding Dp/p = 0.034 per 1°C (Dp is 100, p was taken as the middle value between 200 and 300: 100 ppm/250 ppm = 40% pressure increase per 11.6°C = 3.4%/°C); the second peak from left yielded 12.9%/1°C, the remaining 15 peaks yielded between 2.1 and 4.9 %/°C; the mean value of said 15 peaks was 3.2 (the second peak was omitted, the mean value including it would be 3.8). So the Antarctic ice yielded:
Dp/p = 0.032 per 1°C          (IX)
For the present level of 400 ppm of CO2, it would provide (400 ppm * 0.032 = 12.8):
Dp/DT = 12.8 ppm per 1°C .

Four different (even though not entirely independent) attitudes provided four rough estimations of a CO2 Dp increase due to the release from the oceans for a temperature increases of one degree: 8.7, 7.3, 10.8, 12.8 ppm. In view of the source heterogeneity and gross simplifications in deriving the values, they look reasonably similar. Too low CO2 concentrations during ice ages, bordering on the minimal acceptable amounts for photosynthesis, raises a question of a possible systematic error which might have lowered the experimentally found values in the ice by several tens ppm versus real atmospheric contents in the supposed period. For example, if real values in the third peak should have comprised a CO2 increase from 250 to 350 ppm (instead of 200 to 300), the pressure increase would be 33% instead of 40%, lowering the result by a factor of (33/40=) 0.83, and Dp/p value would be 10.6 per 1°C instead of 12.8 – still closer to the three other estimations. In any case, the average of the four estimations (9.9 ppm per deg) provides increasing the atmospheric level of CO2 due to the lowered solubility of about 10 ppm per 1°C.

The greenhouse effect of CO2 on the surface temperature is often related to in the terms of climate sensitivity, expressed as surface temperature increase a for doubled CO2 concentration in the atmosphere (the temperature is supposed to increase logarithmically with the concentration), the sensitivity to CO2 doubling a being mostly assessed between 0.3 and 3°C, even though the UN uses estimations of up to 6°C. So that doubling of CO2 level from current 400 ppm to 800 ppm (supposed to happen if most known fossil fuel is burnt) would lead to a temperature increase of up to about 1-2°C. Nevertheless, let us continue to consider the worst scenario, and let us consider a warming of 4°C, resulting in releasing (Dp/p/°C = 10) 40 ppm CO2 from the oceans. What greenhouse warming would this increment cause? Let us suppose that a greenhouse temperature increase T1 will be, in a limited range, proportional to the CO2 pressure-increase, as follows:
DTi  =  a * Dp/p          (X)
(for doubling Dp from 400 ppm to 800 ppm, the formula provides DT1 of 400/400*a = a, which is true; of course, the linear interpolation for smaller Dp provides only rough DTi values). Let us now assess the members in the events chain:                           
Dp1 is 400 ppm
DT1 = [from (X)] a * Dp1/p1 = a * 400/400 = a = 4°C
Dp2 =  [from (VII)]  e /T2 * p2 * DT1 = [from (VIII)] 0.027*p2*DT1 = 0.027 * 800 * 4 = 86 ppm
DT2 = a * Dp2/p2 = a * 86/800 = 0.43°C
Dp3 = 0.027*p3*DT2 = 0.027 * 886 * 0.43 = 10.3 ppm
DT3 = a * Dp3/p3 = a * 10.3/886 = 0.047°C
Dp4 = 0.027*p4*DT3 = 0.027 * 896 * 0.047 = 1.14 ppm
DT4 = a * Dp4/p4 = a * 1.14/896 = 0.0051°C
Dp5 = 0.027*p5*DT4 = 0.027 * 897 * 0.0051 = 0.12 ppm
DT5 = a * Dp5/p5 = a * 0.12/897 = 0.00054°C
It can be seen that each pressure increment in the feedback sequence is about ten times lower than the previous one, and the same holds for the temperature increments. Let us estimate the ratio of two following members in the sequence. It holds that  Dpi = 0.027*pi*DTi-1   and   DTi-1 = Dpi-1 * a/pi-1 , so that  for ratio Dpi / Dpi-1 :
Dpi/Dpi-1  =  0.027 *a * pi/pi-1          (XI)
The pressures pi form sequence: 400, 800, 886, 896, 897, which seems to converge to a finite value, so that pi/pi-1 = 1 for higher members, and we can write for higher members:
Dpi/Dpi-1  =  0.027 * a          (XII)
The number 0.027 comes from equation (VII) and equals to e /T2 , when e = 2400 and T = 298. In a narrow range of T, the value e /T2 can be roughly taken as a constant b, characterizing the solubility of CO2 in water. Higher members of the sequence behave like a geometric sequence with a quotient of a*b . The temperature increments of the sequence behave the same way:
DTi/DTi-1  =  0.027*a = a*b          (XIII)
Both geometric sequences have a final sum, since a*b is less than 1. This is expected, because no catastrophic overheating and explosive (runaway) release of CO2 from the oceans has occurred during the Earth history, so that the positive feedback is weak. For sensitivity of about 4°C, a*b is about (4*0.027=) 0.1, and the sum of all Dp increments seems to be about 900 ppm for the initial Dp1 being 400 ppm, so that a human-caused release of, for example, 400 ppm would result in releasing 100 ppm more CO2 from the ocean due to the increased greenhouse effect and lowered CO2 solubility (instead of 800 ppm, we would eventually observe 900 ppm); the positive feedback would increase the warming greenhouse effect by about 0.5°C (the sum of DT2+DT3+DT4+DT5+…), in addition to the supposed 4°K. Shortly, even if having neglected the CO2 dissolution and other cooling factors, and having taken the most severe assumptions in our assessment, the eventual positive feedback caused by reduced CO2 solubility would enhance the effects by about 10% (10% more CO2 and 10% higher DT). We have considered the sensitivity to be 4°K, whereas most of reliable climatologists believe that the value is less than 2°C, such as between 0.5 and 1.5°C; those more realistic sensitivity values would yield the proportionally lower feedback enhancement, such as up to 5% or about 2.5%.

Generally, the greenhouse warming DT including the maximal possible feedback enhancement for doubled CO2 level can be written as the sum of an infinite geometric sequence (the sum of a sequence equals a1/(1-q) if the first member is a1 and quotient is q): 
DT  =  DT1 + DT2 + DT3 + DT4 +      =    DT1/(1 – ab) = a/(1-ab) = [for a=1.5°C] = 1.56°C
The feedback thus enhances the greenhouse warming by 0.06°C for sensitivity 1.5°C, corresponding to 4% enhancement.

Conclusion: The maximal assessment of feedback enhancement of the greenhouse warming due to the lower solubility of CO2 at higher temperatures roughly behaves like an infinite geometric sequence of temperature increments with a quotient a*b, where a characterizes greenhouse warming (and represents temperature sensitivity toward CO2 doubling), and b characterizes the lowered solubility of CO2 at higher temperatures (b = e /T2 , e = dlnH/d(1/T), H is Henry constant, b is about 0.027 at 298°K); the sum of the infinite sequence for sensitivity of 1.5°C is 1.56°C. The positive feedback, due to CO2 lowered solubility, would enhance the greenhouse warming by less than 4% if sensitivity is 1.5°C. Since the consumption of CO2 from the atmosphere by dissolution and other processes has been disregarded in the present assessment, the real feedback would be still lower. The feedback enhancement of the greenhouse warming due to lower solubility of CO2 is negligible.

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