What is the correct value of Climate Sensitivity?

[Bonam]

Looking at your model, why is there a term for the time integral of human emissions, or the time integral of CO2 concentration? These kinds of hysteresis terms would be 2nd order terms, I'd think. To first order, the rate of change of CO2 concentration at any given point in time should just depend on the environmental parameters at that time (emissions, temperature, CO2 concentration).

If you just want a simple unrestricted model, why not just use:

dC(t)/dt = K1 * Emissions(t) + K2 * Temperature(t) + K3 * C(t) + K4

where C(t) is the CO2 concentration at any given point in time, and K1, K2, K3, and K4 are constant. If you're looking for a CO2 decay effect, it should still show up in this model as a negative value of the constant K3 (if you were to solve the differential equation, you'd get an exponential decay term there).

Also, if you use the above model and measure dC(t)/dt and Emissions(t) in the same units, you should get K1 =~ 1, that much seems obvious from basic physics (conservation laws). That would be a good test of your fit, if it gives K1 far off from 1 then it's likely wrong. If you measure C(t) in the units of "total mass of CO2 in the atmosphere" and Emissions(t) in the unit "total mass of CO2 emitted per unit time", that should give you the necessary unit agreement.

[-1=e^ipi]

If the heat sink is glaciers + permafrost + deep ocean, then it's temperature is likely very close to 0 degrees C, and likely to remain static at 0 degrees C even as surface/atmospheric temperatures rise by a few C.

Maybe. But you have to remember that there is far more ocean than ice, and that ice has a larger heat capacity. Are you saying that the justification of approximately 0 C is due to the enthalpy of fusion of ice?

The other thing that is relevant here is that the ocean response is faster than the glacier response and ocean heating would be more relevant to CO2 increase than glaciers.

Looking at your model, why is there a term for the time integral of human emissions, or the time integral of CO2 concentration?

Because I don't know the characteristic temperature of the heat sink (which corresponds to changes in CO2 released from the ocean). So I use the fact that the rate of change of temperature should be proportional to the temperature difference between the surface and the heat sink, then integrate, to get around this.

If you just want a simple unrestricted model, why not just use:

dC(t)/dt = K1 * Emissions(t) + K2 * Temperature(t) + K3 * C(t) + K4

Well the issue here is that it takes longer for the ocean to warm than the surface. Maybe it takes on the order of decades/centuries to catch up. And the warming of the oceans is where you are going to get most of the CO2 emissions due to temperature increase. So by including the other factors it might be possible to estimate the time scale of decay for the oceans, as well as how strong this feedback response is.

But I can try your model and present the results. For the sake of discussion.

[On Guard for Thee]

Already told you, its 42.

[-1=e^ipi]

Okay, so I simplified the model by removing the decay and the constant term on the justification that natural variation does not contribute significantly (on the time scale of a century) to changes in CO2 levels other than through planetary heating. So basically the simplified model only has two mechanisms by which CO2 levels can change; human emissions and emissions due to ocean heating and permafrost melting.

So the model is:

dCO2/dt = (D*CO2(0) – CD*Ts(0)) - D*CO2(t) + E*Human + DE*Integral(0 to t; Human*dt) + CD*T.

Which means I can estimate this by performing the regression:

dCO2/dt = β₀ + β₁*CO2(t) + β₂*Human + β₃*Integral(0 to t; Human*dt) + β₄*T + error

with the restriction β₃/β₂ = -β₁.

So I use the data from 1876 to 2009 to perform the Gauss-Newton estimation and these are my 95% confidence intervals for C, D, E and Ts(1876):

C: -81 +/- 181
D: 0.015 +/- 0.012
E: 0.00038 +/- 0.00011
Ts(1876): 14.14 +/- 0.030

C is the CO2 emitted in ppm due to increasing the characteristic heat sink by 1 celcius. It's negative and very uncertain. Maybe I need to use an empirical value for E to better estimate this.

D is in (year)^-1, and suggests that the decay time of the characteristic heat sink is ~67 years.

E is in ppm per million metric tones. It suggests that about 2.6 billion metric tonnes of carbon is needed to increase atmospheric CO2 by 1 ppm. This seems to be consistent with numbers I see online (which suggests 2-3 billion metric tons).

Ts(1876) corresponds to the characteristic heat sink temperature in celcius for 1876. (note that I am assuming here that global temperatures are the hadcrut4 temperature anomaly data + 14C, so if that assumption is off slightly then this estimate is off by the same amount).

Edited February 13, 2015 by -1=e^ipi

[eyeball]

The trouble is you don't have a clue what the IPCC et. al. actually says.

Sure I do.

Your opinion is based entirely on what Greenpeace and other ideologues tell about what the IPCC says which is quite far from the truth.

You don't know what you're talking about. It's based on all sorts of sources; from print news, TV, Internet (this forum for example), science documentaries, the IPCC website. I'm informed by government publications and notifications to my industry - all manner of sources. I'm pretty sure I've probably scanned something by Greenpeace in all that time but, you're telling me Greenpeace and other ideologues are actually responsible for producing EVERYTHING I've ever read, heard or seen, even government publications? That's just weird.

If you actually looked at the evidence instead bragging about how you are ignorant and proud you might be surprised what it actually says.

I bragged about how I am ignorant....where? You have actual evidence of that or are you just providing an example of how Greenpeace makes crap up to score a point? Don't worry I understand that very clearly when I see it. Like billions of other human beings I know quite clearly what it implies.

[-1=e^ipi]

Maybe. But you have to remember that there is far more ocean than ice, and that ice has a larger heat capacity. Are you saying that the justification of approximately 0 C is due to the enthalpy of fusion of ice?

Actually, I take that back. By the definition I am using, the characteristic temperature of the heat sink is equal to the surface temperature when the Earth is in equilibrium. So at the end of the little ice age when surface temperatures were ~13-14C the heat sink temperature was equal to surface temperature.

[Bonam]

Maybe. But you have to remember that there is far more ocean than ice, and that ice has a larger heat capacity. Are you saying that the justification of approximately 0 C is due to the enthalpy of fusion of ice?

The vast majority of the Earth's ocean water is at a temperature of 0-3 C, hence the approximately 0 C temperature for the heat sink in my earlier statement.

Edited February 13, 2015 by Bonam

[On Guard for Thee]

Sure I do.

You don't know what you're talking about. It's based on all sorts of sources; from print news, TV, Internet (this forum for example), science documentaries, the IPCC website. I'm informed by government publications and notifications to my industry - all manner of sources. I'm pretty sure I've probably scanned something by Greenpeace in all that time but, you're telling me Greenpeace and other ideologues are actually responsible for producing EVERYTHING I've ever read, heard or seen, even government publications? That's just weird.

I bragged about how I am ignorant....where? You have actual evidence of that or are you just providing an example of how Greenpeace makes crap up to score a point? Don't worry I understand that very clearly when I see it. Like billions of other human beings I know quite clearly what it implies.

I suppose they think somebody is actually interested in these Xs and Os games. Thankfully real scientists are actually working on the problem

[-1=e^ipi]

Now that I think about it, I think the model used in post #154 is missing one important factor. And this factor explains why I got a negative and uncertain estimate for the CO2 released by increasing the temperature of the characteristic heat sink of earth.

Increasing the temperature of the characteristic heat sink does cause it to release CO2.

However, if one increases atmospheric CO2 then the oceans will want to absorb some of this extra CO2 in order to get to thermodynamic equilibrium. So having unusually high CO2 levels will cause CO2 to be absorbed over time, which will reduce the rate of increase of atmospheric CO2.

Since the increase in temperatures over the past century are very correlated with increase in CO2, the factor I used in the model in post #154 that describes the increase in rate of atmospheric CO2 increase due to warming the heat sink picked up the decrease in the rate of atmospheric CO2 increase due to oceans absorbing CO2, which made the estimate negative. In fact, it should be expected that the rate of CO2 being absorbed by the ocean since atmospheric CO2 concentrations are higher than normal more than offsets the rate of CO2 being released by the oceans due to the oceans warming as we can see here:

http://www.skepticalscience.com/human-co2-smaller-than-natural-emissions.htm

Since the oceans contain far more carbon than the atmosphere, it is reasonable to approximate the rate at which the oceans absorb CO2 due to having less 'effective CO2' than the atmosphere as being proportional to the difference between atmospheric CO2 and some 'characteristic CO2 ppm of the earth's carbon sink'. So the model (omitting volcanism and other natural causes for reasons I explained earlier) would be:

dCO2/dt = E*Human_emissions(t) + C*dTs/dt + G*dCO2s/dt

Where Ts is the characteristic temperature of the Earth's heat sink

and CO2s is the characteristic CO2 ppm of the Earth's carbon sink.

Plus there would be the two equations (that came from linear approximations):

dTs/dt = D*(T(t) - Ts(t))

and dCO2s/dt = H*(CO2(t) - CO2s(t))

Despite not knowing Ts(t) and CO2s(t), I could eliminate them by taking the differential equation + the linear approximations and integrating like I did earlier.

However, the problem is that I would have added 3 unknowns (G, H, CO2s(0)), but I would not have added any explanatory variables since I am already using CO2 in my regression. The regression in post #154 has 5 explanatory variables and 4 unknowns. A new regression would have 5 explanatory variables and 7 unknowns. So I would need to use a priori information to reduces the number of unknowns to 5 in order to be able to estimate it.

I could assume that the Earth's temperature sink and carbon sink were in equilibrium in ~1850 (which might be reasonable).

Alternatively, I could assume that the decay rate of the Earth's carbon sink and the decay rate of the Earth's temperature sink are the same (which might be the case for example if they are both driven primarily by ocean convection as opposed to diffusion).

In any case, I'll think about making modifications to the model in post #154.

[-1=e^ipi]

Also, if the approximation that the decay rate for the Earth's carbon sink is equal to the decay rate of the Earth's temperature sink holds, then the decay rate of ~67 years is valid. Note that this does seem consistent with the decay rate I got using the IPCC information and posted in the original post (~63.3 years). Interestingly, there are many climate cycles that have this time scale as well (such as the North Atlantic Multidecadal oscillation, the oscillation in the length of the day, and perhaps solar cycles such as the Gleissburg cycle).

So if we think about how the Earth responds to increases in atmospheric CO2, we can think about it on 3 different time scales.

In the short run (with the decay time of ~1 year), one gets a fast response due to the direct forcing effects of CO2 and the fast feedback effects of water vapour and lapse rate. This gives the transient climate sensitivity (not to be confused with the transient climate response).

In the medium run (with a decay time of 60-70 years), one gets an additional response as the oceans & permafrost warm and obtain equilibrium with the increase in surface temperatures due to the CO2. This gives the equilibrium climate sensitivity.

And in the very long run (with a decay time of ~1000 years), one gets an additional response due to glaciers melting and vegetation changing (which gives the Earth system sensitivity). Of course since this timescale is less than the timescale of excess CO2 being absorbed by the oceans, the Earth system sensitivity will not get released, so the equilibrium climate sensitivity is more relevant.

If this is the case, then the climate sensitivity value I got in post #92 ((1.98 +/- 0.92) Celcius) is closer to the transient climate sensitivity than the equilibrium climate sensitivity since it came with a decay time of ~3 years.

If the actual decay time to the equilibrium response is ~63 years, then my value in post #92 is underestimating equilibrium climate sensitivity by a factor approximately (1 - 1/63)/(1 - 1/3) = 1.48. So this means that the results I got in post #92 actually suggest a climate sensitivity of ~3C and are therefore not in contradiction with the results I got from correcting Loehle's paper as well as the mainstream scientific opinion and the IPCC.

So we are looking at a transient climate response of ~2C, and equilibrium climate sensitivity of ~3C, and an earth system sensitivity of 4-4.5C.

Edited February 15, 2015 by -1=e^ipi

[-1=e^ipi]

The vast majority of the Earth's ocean water is at a temperature of 0-3 C, hence the approximately 0 C temperature for the heat sink in my earlier statement.

Maybe think of it as a concept similar to the centre of mass.

The centre of mass in the Pluto - Charon system isn't in either Pluto nor Charon. So the vast majority of the mass of the Pluto - Charon system isn't near the centre of mass. But that doesn't mean the centre of mass of this system isn't a useful concept.

Or maybe ask yourself this: What prevents the water at the bottom of the ocean from being at the same temperature as the surface of the ocean? This isn't as simple as saying colder water is heavier because then you have to explain why some water is colder than others and what prevents heat transfer from the surface to the deep ocean until isothermal conditions are reached.

Edited February 15, 2015 by -1=e^ipi

[-1=e^ipi]

If you just want a simple unrestricted model, why not just use:

dC(t)/dt = K1 * Emissions(t) + K2 * Temperature(t) + K3 * C(t) + K4

where C(t) is the CO2 concentration at any given point in time, and K1, K2, K3, and K4 are constant. If you're looking for a CO2 decay effect, it should still show up in this model as a negative value of the constant K3 (if you were to solve the differential equation, you'd get an exponential decay term there).

So I estimated this using 1876-2009 data and I get the following 95% confidence intervals:

K1 = (-1.2 +/- 2.7) ppm per celcius

K2 = (4.0 +/- 1.2)x10^-4 ppm per million tons of carbon

K3 = (-1.54 +/- 1.18)x10^-2 per year

K4 = (4.6 +/- 3.4) ppm per year

Make of that what you wish. So I guess this model assumes that oceans respond immediately to temperature changes, but assumes that the CO2 decay effect is not negligibly small? If I were to assume that this model's assumptions are valid then it suggests a decay time of excess CO2 of ~65 years. If this is primarily due to interact with the oceans, then the fact that it is similar to the decay time of ~63 years in post #1 and the decay time of ~67 years in post #154 is interesting.

Also, if you use the above model and measure dC(t)/dt and Emissions(t) in the same units, you should get K1 =~ 1, that much seems obvious from basic physics (conservation laws). That would be a good test of your fit, if it gives K1 far off from 1 then it's likely wrong. If you measure C(t) in the units of "total mass of CO2 in the atmosphere" and Emissions(t) in the unit "total mass of CO2 emitted per unit time", that should give you the necessary unit agreement.

My understanding is that there is a bit of uncertainty about how many millions of metric tons of carbon are needed to increase atmospheric CO2 by 1 ppm. The results of the regression I just did suggests that ~2.5 billion metric tons of carbon are needed, which agrees with the 2.6 billion metric tones I got in post #154, and also with the range of estimates in the scientific literature.

Edited February 17, 2015 by -1=e^ipi

[-1=e^ipi]

I thought I would try to get a rough idea of what the timescale of the fast response (due to direct heating of the earth’s surface from increased CO2, as well as perhaps the water vapour feedback) is in order to get an idea of how this compares to the 60-70 year ‘ocean’ response time.

Consider a representative square meter of the Earth’s surface at the global average surface temperature of 15 C and with a relative humidity of 75% (note that if you look at relative humidity across the Earth’s surface, it does not have much temperature dependence).

First, let’s try to estimate the ‘fast’ heat capacity of this representative square meter. To do this I’ll use all of the air above the surface and the first 10 metres of ground/water below the surface. I justify the choice of 10 meters on the basis that this seems to be roughly the characteristic depth at which temperature changes rapidly to surface temperature (although I might be being generous; anyway, this is a rough calculation).

The standard atmospheric pressure at sea level is 101325 Pascals. Since the acceleration due to gravity at the Earth’s surface is roughly 9.81 m/s², this means there is roughly 10329 kg of air above this square metre of surface.

Dry ‘air’ (78% N₂, 21% O₂ + other stuff) has a specific heat capacity of roughly 1006 J/kg/C at atmospheric pressure. However, the air isn’t dry. If we assume that all of this air is at 15 C and has a relative humidity of 75%, then by the clasius-clapeyron relation (specifically the August-Roche-Magnus formula, see http://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation for more details), such air would be 1.26% water vapour. The specific heat capacity of water vapour is roughly 1864 J/kg/C. The specific heat capacity of the 75% relative humid air is the weighted average, so is 1017 J/kg/C.

However, if one were to maintain the 75% relative humidity and increase temperature, then one needs to take into account the fact that water vapour needs to be created by vaporizing water. The enthalpy of vaporization of water is roughly 2260 kJ/kg. If one takes the first derivative of the clasius-clapeyron relation then one obtains that at 15 C and atmospheric pressure that the increase in the percentage of air that is water that is required to maintain 75% relative humidity if one increases temperature by 1 C is 0.081%. This means that about 1831 J/kg/C are needed to maintain the relative humidity. This means that the effective specific heat capacity of the air is ~ 2848 J/kg/C. So that 10329 kg of humid air has an effective heat capacity of approximately 29417 kJ/C.

Now let’s consider that 10 m of soil/water below the surface. The density of fresh water is 1000 kg/m³, the density of sea water is 1027 kg/m³, and the density of soil varies between 1000 kg/m³ and 1600 kg/m³. Furthermore, the specific heat capacity of soil is roughly 1480-2090 J/kg/C and the specific heat capacity of fresh water is 4182 J/kg/C. The soil has a higher density but lower heat capacity (and a lower heat capacity times density). To be on the safe side, I’ll round up slightly and assume that we have 10 m of fresh water below the surface. This corresponds to 10000 kg of water directly below the surface, which has a heat capacity of 41820 kJ/C.

So the combined heat capacity that corresponds to this square meter of the Earth’s surface is ~71237 kJ/C.

Assume (for simplicity) that the 10m of water underneath the surface and the air above the surface has a uniform temperature of 15C and radiates as a black body. Then the power of the radiation emitted by this square metre is m²*σT⁴, where σ is the Stefan-Boltzmann constant (5.6704 x 10^-8Wm^-2K^-4). To get the power of radiation emitted to space, divide this by two (since half of the radiation will go upwards, and half will go downwards).

In equilibrium, the blackbody radiation emitted must equal incoming radiation (from the sun, due to greenhouse gases, etc.), which we shall call F, the forcing. We get F = m²*σT⁴/2. Now consider small changes in the forcing such that we can use a taylor approximation to get ΔF = 2m²*σ T³ΔT, where ΔF and ΔT is the change in forcing and equilibrium temperature respectively.

If we start in equilibrium and increase by a small amount ΔF, the temperature will slowly decay (exponentially) towards the new equilibrium from the old equilibrium. Therefore we can write the temperature as a function of time as T(t) = T(0) + ΔT*(1 - exp(-βt)), where t=0 corresponds to the time where there is the sudden change in forcing, and β is a parameter that determines the rate of decay. Taking the first derivative of this with respect to time gives dT(t)/dt = βΔT*exp(-βt). If we set time equals zero, we get dT(0)/dt = βΔT.

Now we know that at t=0, the difference between outgoing and incoming radiation is ΔF, since right before t=0 things were at equilibrium. Given that the heat capacity is 41820 kJ/C, one can get the change in heat per unit time at t=0 as dT(0)/dt = ΔF/(41820 kJ/C). However, dT(0)/dt = βΔT and ΔF = 2m²*σ T³ΔT. Thus we get 2m²*σ T³ΔT/(41820 kJ/C) = βΔT.

=> β = 2m²*σ T³/(41820 kJ/C). Using T = (273.15 + 15) K, one obtains: β = 6.488 x 10^-8 s^-1

Inverting this gives the decay time, which is 15412904 seconds = 178.4 days = 0.488 years.

Therefore, the fast decay time is roughly half a year, which is far less than the ocean decay time of 60-70 years. Of course, with water vapour the fast decay time might be slightly longer (since it would take time for water vapour to get into the air, then increase forcing, which causes more water vapour), though I may have been generous in my estimation of heat capacity.

In any case, if we go with a CO2 + water vapour equilibrium climate response of 1.74 C (this is roughly the fast response that neglects various other feedback effects that would occur on longer time scales). Then after doubling CO2, within a year temperatures should increase to within 1/e^2 of this value (by about 1.5 C).

Edited February 17, 2015 by -1=e^ipi

[-1=e^ipi]

dCO2/dt = E*Human_emissions(t) + C*dTs/dt + G*dCO2s/dt

Where Ts is the characteristic temperature of the Earth's heat sink

and CO2s is the characteristic CO2 ppm of the Earth's carbon sink.

Plus there would be the two equations (that came from linear approximations):

dTs/dt = D*(T(t) - Ts(t))

and dCO2s/dt = H*(CO2(t) - CO2s(t))

I thought I would try to solve this system of equations, to get something that I can estimate via a regression. For simplicity of notation, let Human be the CO2 emissions due to humans, IHuman be the integral of Human, IIHuman be the integral of IHuman, IT be the integral of T, and ICO2 be the integral of atmospheric CO2 (otherwise notation is the same as I previously had).

Start with the equations:

dCO2/dt = E*Human(t) + C*dTs/dt + G*dCO2s/dt (1)

dTs/dt = D*(T(t) - Ts(t)) (2)

dCO2s/dt = H*(CO2(t) - CO2s(t)) (3)

Substituting (3) into (1) gives:

dCO2/dt = E*Human(t) + C*dTs/dt + GH*(CO2(t) - CO2s(t)) (4)

Integrating (1) with respect to time gives:

CO2(t) – CO2(0) = E*IHuman(t) + C*(Ts(t) – Ts(0)) + G*CO2s(t) – G*CO2s(0)

= E*IHuman(t) - C*(T(t) – Ts(t) + Ts(0) – T(t)) + G*CO2s(t) – G*CO2s(0) (5)

Substituting (1) into (5) gives:

CO2(t) – CO2(0) = E*IHuman(t) – C/D*dTs/dt + C*(T(t) – Ts(0)) + G*CO2s(t) – G*CO2s(0) (6)

H*(6) + (4):

dCO2/dt + H*CO2(t) – H*CO2(0) = E*Human(t) + C(1-H/D)*dTs/dt + GH*CO2(t) + EH*IHuman(t) – GH*CO2s(0) + CH*(T(t) – Ts(0))

=> dCO2/dt = E*Human(t) + C(1-H/D)*dTs/dt + H(G-1)*CO2(t) + EH*IHuman(t) + CH*T(t) – CH*Ts(0) – GH*CO2s(0) + H*CO2(0) (7)

Substituting (2) into (7) gives:

dCO2/dt = E*Human(t) + C(D-H)*(T(t) - Ts(t)) + H(G-1)*CO2(t) + EH*IHuman(t) + CH*T(t) – CH*Ts(0) – GH*CO2s(0) + H*CO2(0) (8)

Integrating (7) gives:

CO2(t) – CO2(0) = E*IHuman(t) + C(1-H/D)*(Ts(t) – Ts(0)) + H(G-1)*ICO2(t) + EH*IIHuman(t) + CH*IT(t) + (– CH*Ts(0) – GH*CO2s(0) + H*CO2(0))*t (9)

(8) + D*(9):

dCO2/dt + D*CO2(t) – D*CO2(0) = E*Human(t) + C(D-H)*(T(t) - Ts(t)) + H(G-1)*CO2(t) + EH*IHuman(t) + CH*T(t) – CH*Ts(0) – GH*CO2s(0) + H*CO2(0) + ED*IHuman(t) + C(D-H)*(Ts(t) – Ts(0)) + HD(G-1)*ICO2(t) + EDH*IIHuman(t) + CDH*IT(t) + DH(– C*Ts(0) – G*CO2s(0) + CO2(0))*t

=> dCO2/dt = [(D+H)*CO2(0) – GH*CO2s(0) – CD*Ts(0)] + E*Human(t) + E(H+D)*IHuman(t) + EDH*IIHuman(t) +CD*T(t) + CDH*IT(t) + (H(G-1)-D)*CO2(t) + HD(G-1)*ICO2(t) + DH(– C*Ts(0) – G*CO2s(0) + CO2(0))*t;

Explanatory Variables are: constant, Human, IHuman, IIHuman, T, IT, CO2, ICO2, t = 9 explanatory variables.

Parameters are: E, C, G, D, H, Ts(0), CO2s(0) = 7 parameters

So it is possible to estimate this.

Edited February 17, 2015 by -1=e^ipi

[On Guard for Thee]

Its 42.

[-1=e^ipi]

Sorry, I made a mistake in my last post. It should be corrected now.

[-1=e^ipi]

I just noticed that I made an error in the code that I used to get the results for post #154. I miscalculated an integral. Below is the corrected 95% confidence intervals.

C: -26 +/- 62
D: 0.043 +/- 0.026
E: 0.000257 +/- 0.000009
Ts(1876): 14.34 +/- 0.79

C is the CO2 emitted in ppm due to increasing the characteristic heat sink by 1 celcius. It's still negative and not statistically significant.

D is in (year)^-1, and suggests that the decay time of the characteristic heat sink is 23 years. I guess getting it in the 60-70 range a few posts ago was a coincidence. In any case, this model is not accurate for reasons I explained in post #159.

E is in ppm per million metric tones. It suggests that about 3.9 billion metric tonnes of carbon is needed to increase atmospheric CO2 by 1 ppm.

Ts(1876) corresponds to the characteristic heat sink temperature in celcius for 1876.

Edited February 17, 2015 by -1=e^ipi

[-1=e^ipi]

I've attempted a Gauss-Newton method estimate for the model described in post #164.

Recall that the model was essentially the following 3 equations.

dCO2/dt = E*Human(t) + C*dTs/dt + G*dCO2s/dt

dTs/dt = D*(T(t) - Ts(t))

dCO2s/dt = H*(CO2(t) - CO2s(t))

The 95% confidence intervals for the estimates are:

E: 0.00034 +/- 0.00018

C: 51 +/- 97

D: 0.35 +/- 0.63

G: -0.0027 +/- 0.0047

H: 0.068 +/- 0.090

Ts(1876): 13.97 +/- 0.06

CO2s(1876): 271 +/- 54

E is in ppm per million metric tones. It suggests that about 2.9 billion metric tonnes of carbon is needed to increase atmospheric CO2 by 1 ppm.

Ts and CO2s represent the temperature of the characteristic heat sink and the CO2 concentration of the characteristic carbon sink in Celcius and ppm respectively. They are reasonable values.

C, D, G, F are all unreasonable and not significant. As expected, the increase in CO2 concentrations due to the warming of the oceans is very correlated with the decrease in CO2 concentrations due to the atmosphere having excess CO2 relative to the ocean. Either I should try to get better data (either use all the data back to 1850 or use monthly data post 1950), or I should try to impose some reasonable conditions on the values of C and D.

Edited February 19, 2015 by -1=e^ipi

[-1=e^ipi]

Chemistry is not my strongest area, but as I understand it, Henry's constant determines the ratio between the solubility of gas in a liquid and the partial pressure of the gas above that liquid.

http://en.wikipedia.org/wiki/Henry%27s_law#Temperature_dependence_of_the_Henry_constant

From the wiki, one gets that the solubility of CO2 in water is proportional to exp(C*(1/T + 1/(298K))), where T is the temperature of the water in Kelvin and C = 2400K for CO2.

Taking the natural logarithm of this gives C*(1/T + 1/(298K)). Plotting this from 0 C to 30 C gives a line of best fit of -0.0126 +/- 7.3095 and gives a very good fit (R^2 is 0.9993). Thus the amount of CO2 in water should decrease by about 1.26% if the temperature increases by 1 Celcius.

Now the oceans contain around 38000 billion tons of carbon (http://worldoceanreview.com/en/wor-1/ocean-chemistry/co2-reservoir/). So if the temperature of the ocean were to increase uniformly by 1 Celcius, then this would release ~47.88 billion tons of carbon. If the value that 2.9 billion tons of carbon is needed to increase atmospheric CO2 by 1 ppm is correct (see post #168), then increasing the ocean temperature uniformly by 1 degree celcius should increase atmospheric CO2 by ~16.5 ppm.

So the value of C in the last 2 regressions should be ~16.5. This is not a very large effect compared to Human emissions and to CO2 being absorbed by the ocean due to imbalance between CO2 concentrations in the Ocean vs CO2 concentrations in the atmosphere.

[-1=e^ipi]

I guess I need to consider permafrost as well, otherwise the 16.5 value in my last post is an underestimate.

There are 1400-1700 billion tons of carbon in permafrost (http://en.wikipedia.org/wiki/Permafrost); I'll use the central estimate of 1550 billion tons. Due to lack additional a priori knowledge, I'll assume that the permafrost has a similar behaviour as the oceans (so 1 degree celcius increase releases ~1.26% of the carbon). Also, permafrost is located primarily in polar regions, so I should take into account polar amplification. The global polar amplification factor is ~2.0 (note that in the jetstream thread, I eventually used a polar amplification of 2.5; the reason was because I was considering the northern hemisphere, which has a higher polar amplification factor than the southern hemisphere). Putting these factors together (plus the assumption of 2.9 billion tons of carbon corresponding to 1 ppm) suggests that I should increase the 16.5 by ~13.5 to ~30.

Note that if the equilibrium climate sensitivity is approximately 3 celcius then this suggests that doubling CO2 concentrations should release an additional 3*30 = 90 ppm of CO2 in the long run (due to ocean warming and permafrost melting). Note that this corresponds very well with the approximately 87 ppm value that is the central estimate of the ranges in the IPCC's AR4 (chapter 7).

So the value of C in the previous regressions I did should be ~ 30.

[-1=e^ipi]

Found a much more serious error in my code. I was accidentally using aerosol data instead of temperature data (sorry I mixed up a 6 with a 7). All the regressions since post #150 are nonsense.

I tried the regression of post #168 but with the code correction. The 95% intervals are:

E: 0.0028 +/- 0.0334

C: -61 +/- 549

D: 1.7 +/- 5.0

G: -0.0061 +/- 0.0421

H: 0.068 +/- 3.101

Ts(1876): 14.07 +/- 3.96

CO2s(1876): 272 +/- 115

Nothing is very significant; probably because the model is too complex for the data. I'll have to impose some restrictions (using a priori information of course).

Edited February 19, 2015 by -1=e^ipi

[cybercoma]

Can someone let me know when he monkeys around with the formulas long enough that he gets back to the established record?

[-1=e^ipi]

dC(t)/dt = K1 * Emissions(t) + K2 * Temperature(t) + K3 * C(t) + K4

So I re-estimated Bonam's model using 1876-2009 data (sorry about the temperature mistake) and I get the following 95% confidence intervals:

K1 = (0.00039 +/- 0.00013) ppm per celcius

K2 = (-0.058 +/- 0.428) ppm per million tons of carbon

K3 = (-0.014 +/- 0.014) per year

K4 = (4.3 +/- 4.2) ppm per year

Make of that what you wish. If I were to assume that this model's assumptions are valid then this suggests that 2.6 billion tons of carbon is needed to increase atmospheric CO2 by 1 ppm and that atmospheric CO2 has a decay time of approximately 70 years.

[-1=e^ipi]

Let's assume that the ~87 ppm CO2 feedback effect for a doubling of CO2 as the IPCC's central estimate is correct (posts 169-170 give justification for this). Furthermore, let's suppose that equilibrium climate sensitivity is 3 C and the carbon sink decays by 1.26% if its temperature is increased by 1 C.

Then 87 = C*Integral(T=0 to 3; exp(-0.0126*T)dT) = -C/0.0126*(exp(-0.0126*3)-1)

=> C = 29.55, where C is the constant I have referred to in the last few posts that represents the increase in atmospheric CO2 due to increasing the heat sink by 1 degree.

I'll assume that C = 29.55, because it is clear that my model in post #171 cannot distinguish between change in CO2 levels due to changes in the temperature of the heat sink and changes in CO2 levels due to changes in the carbon sink (since temperature and CO2 trends are very correlated).

If I perform the regression of post #171, but fix C = 29.55, and CO2s(1876) = 275 (roughly pre-industrial levels) then my 95% confidence intervals are:

E: 0.00034 +/- 0.00019

D: -0.61 +/- 0.72

G: -0.002 +/- 0.015

H: 0.013 +/- 0.005

Ts(1876): 12.7 +/- 8.2

Not bad, but I might need to use more a priori information for additional restrictions to get more reasonable results (especially since the effect of warming the heat sink on CO2 concentrations is so small).

This does suggest that it takes ~ 2.9 billion tons of carbon to increase atmospheric CO2 by 1 ppm and that the decay time of the carbon sink towards equilibrium is ~75 years (comparable to the estimate in Bonam's model).

I also checked the residual. I still see no evidence of variation in change CO2 concentrations due to volcanism.

Edited February 18, 2015 by -1=e^ipi

[Bonam]

I wouldn't really call what I posted above "my model"... it's just a very basic linear approximation.

Almost any non-linear function can be approximated as linear over a sufficiently small range. Therefore, for small changes, assuming that all the factors are linear as I did to write that equation can be an adequate representation. For typical types of functions that one may encounter in many physical situations, like exponentials and power laws, etc, the linear approximation might perhaps be sufficiently accurate for +/- 10% around a given value. However, the approximation would no longer be valid for a doubling of the value. Therefore, the model would be of limited/no use in trying to predict what would happen under a doubling of CO2.

To try to make a model that is accurate through such a large extrapolation would require correctly capturing all the relevant physics, including any non-linear terms, as you've tried to do, though it sounds like you probably still have some work left (this is, after all, what hundreds of scientists are spending their careers doing). If one captures all the relevant physics (zeroth and 1st order effects), the model should fit to the data with error bars on your coefficients of only a few %, in my experience.

Have you graphed temperature and CO2 levels as a function of time as calculated by your model? How well do they overlay your historical time series for these values from 1876-2009?

Edited February 18, 2015 by Bonam