<p>To get a rough estimate you should be using a logarithmic interpolation rather than a linear interpolation. The response to added <span class="math-container">$\text{CO}_2$</span> would be linear if the atmosphere had barely any <span class="math-container">$\text{CO}_2$</span> in it. Even 280 ppm does not qualify as "very little". "Very little" would mean a handful of parts per million, at most.</p>

<p>The issue is that the atmosphere is rather opaque at those frequencies where <span class="math-container">$\text{CO}_2$</span> absorbs and emits thermal infrared radiation. Instead of a linear response, the response in terms of radiative forcing is roughly logarithmic. This is why climatologists talk about the effect of a doubling the <span class="math-container">$\text{CO}_2$</span> levels. The radiative forcing of an increase from 140 ppm <span class="math-container">$\text{CO}_2$</span> to 280 ppm is more or less the same as the response to an increase from 280 ppm to 560 ppm, or to an increase from 560 ppm to 1120 ppm.</p>

CC BY-SA 4.0
2019-10-29T06:01:11.380Z
"To get a rough estimate you should be using a logarithmic interpolation rather than a linear interpolation." Yes that is where I thought I was probably wrong. How do we do this?
polcott
CC BY-SA 4.0
2019-10-29T06:05:21.987Z
@polcott - Assuming a 3 °C increase due to a doubling of $\text{CO}_2$ level, the response to a 73% increase (for example) is $3\log_2(1.73) \approx 2.37$, where $\log_2$ is the base 2 logarithm function.
David Hammen
CC BY-SA 4.0
2019-10-29T23:47:35.493Z
How to we account for 2.37C being greater than the reported < 1.5C ?
polcott
CC BY-SA 4.0
2019-10-30T02:26:56.250Z
@polcott - Because the climate has not yet caught up to that increase.
David Hammen
CC BY-SA 4.0
2019-10-30T05:32:39.280Z
I was hoping that I calculated it incorrectly. If my calculations are correct then those of IPCC is wrong or are they only referring to 1.5C as the near term or immediate term feedbacks?
polcott
CC BY-SA 4.0
2019-10-30T09:56:18.957Z
Suppose that over the next 24 hours humanity dumps another 140 ppm of $\text{C}_2$ into the atmosphere, and then only emits just enough $\text{CO}_2$ to keep it steady at 560 ppm. The climate will not suddenly change overnight to be 3°C warmer than preindustrial times. It will take several decades for the climate to fully respond to that step change. Climate changes rather slowly. The huge amount of $\text{CO}_2$ humanity has added to the atmosphere over the last 50 years is essentially a step change to which the climate has.not yet fully responded.
David Hammen
CC BY-SA 4.0
2019-10-30T10:11:01.103Z
I was unthinkingly using your 73% value from another question you asked. That would correspond to 484 ppm of $\text{CO}_2$. Using 410 ppm (which is what you used in that other question), the correct math is that one would expect a $3\log_2\left(\frac{410}{280}\right)$ degree increase, or about 1.65°C.
David Hammen
CC BY-SA 4.0
2019-10-30T14:11:34.827Z
That is still more than the claimed: "we are not yet up to 1.5C yet" claim. The transient response is defined by scientists as the temperature response over human time scales of around 70 years, the equilibrium climate sensitivity over centuries, and finally the Earth system sensitivity after multiple millennia.

https://en.wikipedia.org/wiki/Climate_sensitivity#Different_forms_of_climate_sensitivity
polcott
CC BY-SA 4.0
2019-10-30T22:29:08.010Z
I just found this. FIGURE 1.5
Warming commitment from past emissions of greenhouse gases and aerosols https://www.ipcc.ch/site/assets/uploads/sites/2/2019/01/figure-5-pdf-e1547458812152.jpg It seems to indicate that IPCC said we already have 1.5C that is on its way. I have to study it more and more to be sure.
polcott
CC BY-SA 4.0
2019-10-30T22:58:56.673Z
3 * log2(396 / 280) = 1.5C. 396 PPM would be 2013 data.
polcott
CC BY-SA 4.0
2019-10-29T05:56:16.107Z
2019-10-29T05:56:16.107Z
David Hammen
<p>You can use the formula ΔT=λ·α·㏑(C/C₀), (natural logarithm) where C is the later concentration of CO₂ and C₀ the earlier concentration. λ and α is constants. This formula is valid for a solid sphere but since 70 % of the surface of Earth is water, which warms up much deeper and accumulate about 90 % of the accumulated energy, it isn't totally clear how to use the formula correct.</p>

<p>One way could be to use the formula on global land temperatures and transform to global temperature for sea and land afterwards, somehow.</p>

<p>Otherwise, the calculations won't be exact.</p>

CC BY-SA 4.0
2019-12-27T18:31:51.980Z
2019-12-27T19:57:42.257Z
2019-12-27T19:57:42.257Z
Lehs
<h1>How do we allocate partial CO2 doublings for Empirical Climate Sensitivity? I applied partial doublings on a linear basis, is this correct?</h1>
<p>Climate Threat to the Planet:Implications for Energy Policy and Intergenerational Justice Jim Hansen December 17, 2008
<a href="http://www.columbia.edu/%7Ejeh1/2008/AGUBjerknes20081217.pdf" rel="nofollow noreferrer">http://www.columbia.edu/~jeh1/2008/AGUBjerknes20081217.pdf</a></p>
<p><strong>Empirical Climate Sensitivity 3 ± 0.5C for 2 x CO2</strong><br>
(Hanson said it could be 6C in the longer term)<br>
<a href="http://www.columbia.edu/%7Ejeh1/2008/TargetCO2_20080407.pdf" rel="nofollow noreferrer">http://www.columbia.edu/~jeh1/2008/TargetCO2_20080407.pdf</a></p>
<p>If we assume that the formula above still holds and<br>
assume the current rate of CO2 increasing at 1.9% per year<br>
and assume 280 PPM as the CO2 base. <br>
and assume that partial doublings can be allocated linearly. <br></p>
<p>Years.......CO2<br>
From.......Per.......CO2<br>
Now........Year......Total<br>
....0..........3.00......410<br>
....4..........3.17......420.....1.5C<br>
...17.........4.05......467.....2.0C<br>
...27.........4.89......513.....2.5C<br>
...36.........5.80......560.....3.0C<br></p>
<p>Keith McClary provided this link, which provided the data below it:<br>
<a href="https://en.wikipedia.org/wiki/Climate_sensitivity#Estimating_climate_sensitivity" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Climate_sensitivity#Estimating_climate_sensitivity</a></p>
<p><strong>IPCC Fifth Assessment Report 2014</strong><br>
<strong>66% likely: Empirical Climate Sensitivity 3 ± 1.5C for 2 x CO2</strong><br>
<strong>Possible: Empirical Climate Sensitivity 1C to 6C for 2 x CO2</strong><br></p>
<h1>The best information available provided above shows the climate sensitivity. The next thing that we need is the procedure for allocating fractional doubling.</h1>
<p><a href="https://www.ipcc.ch/site/assets/uploads/sites/2/2019/05/SR15_Chapter1_Low_Res.pdf" rel="nofollow noreferrer">https://www.ipcc.ch/site/assets/uploads/sites/2/2019/05/SR15_Chapter1_Low_Res.pdf</a>
<strong>SPECIAL REPORT: GLOBAL WARMING OF 1.5 ºC<br>
CH 01<br>
Framing and Context</strong>  The following is on page 66<br>
Expert judgement based on the available evidence (including model simulations, radiative forcing and climate sensitivity) suggests that if all anthropogenic emissions were reduced to zero immediately, any further warming beyond the 1°C already experienced would likely be less than 0.5°C over the next two to three decades, and also likely less than 0.5°C on a century time scale.</p>
<h1>It seems we already missed the 1.5C target:<br> 3 * log2(422/280) degree increase, or about 1.78°C</h1>

CC BY-SA 4.0
2019-10-21T11:10:11.223Z
Sorry, non-native speaker here, what do you mean with 'allocate [] doublings'? Is it  also possible that you mean not 'doubling' but simply 'increase'? You might get a quicker response if you simplify the language a bit without dumbing it down.
AtmosphericPrisonEscape
CC BY-SA 4.0
2019-10-21T14:01:48.560Z
@AtmosphericPrisonEscape When CO2 doubles from 280 to 560 we get a 3C increase in global temperature. How much increase do we get when CO2 increases 50% from 280 to 420? I guess 1.5C.
polcott
CC BY-SA 4.0
2019-10-21T22:01:06.477Z
[The radiative forcing, and hence the change in temperature, is proportional to the logarithm of the concentration of infrared-absorbing ("greenhouse") gases in the atmosphere, as quantified by Arrhenius in the 19th century. The sensitivity of temperature to atmospheric gasses, most notably CO
2, is often expressed in terms of the change in temperature per doubling of the concentration of the gas.](https://en.wikipedia.org/wiki/Climate_sensitivity#Estimating_climate_sensitivity)
Keith McClary
CC BY-SA 4.0
2019-10-22T06:34:10.333Z
It is not as simple as that. I will recommend you [this](http://www.realclimate.org/index.php/data-sources/#Rean_output). In fact, David Archer have a few free courses when you can get additional information, including Python models.
David García Bodego
<climate-change>
How do we allocate partial CO2 doublings for Empirical Climate Sensitivity?
polcott
# How do we allocate partial CO2 doublings for Empirical Climate Sensitivity?

(Asked by: username_0 on 2019-10-21)

# How do we allocate partial CO2 doublings for Empirical Climate Sensitivity? I applied partial doublings on a linear basis, is this correct?

Climate Threat to the Planet:Implications for Energy Policy and Intergenerational Justice Jim Hansen December 17, 2008
[http://www.columbia.edu/~jeh1/2008/AGUBjerknes20081217.pdf](http://www.columbia.edu/%7Ejeh1/2008/AGUBjerknes20081217.pdf)

**Empirical Climate Sensitivity 3 ± 0.5C for 2 x CO2**  
(Hanson said it could be 6C in the longer term)  
[http://www.columbia.edu/~jeh1/2008/TargetCO2\_20080407.pdf](http://www.columbia.edu/%7Ejeh1/2008/TargetCO2_20080407.pdf)

If we assume that the formula above still holds and  
assume the current rate of CO2 increasing at 1.9% per year  
and assume 280 PPM as the CO2 base.   
and assume that partial doublings can be allocated linearly.

Years.......CO2  
From.......Per.......CO2  
Now........Year......Total  
....0..........3.00......410  
....4..........3.17......420.....1.5C  
...17.........4.05......467.....2.0C  
...27.........4.89......513.....2.5C  
...36.........5.80......560.....3.0C

username_2 provided this link, which provided the data below it:  
<https://en.wikipedia.org/wiki/Climate_sensitivity#Estimating_climate_sensitivity>

**IPCC Fifth Assessment Report 2014**  
**66% likely: Empirical Climate Sensitivity 3 ± 1.5C for 2 x CO2**  
**Possible: Empirical Climate Sensitivity 1C to 6C for 2 x CO2**

# The best information available provided above shows the climate sensitivity. The next thing that we need is the procedure for allocating fractional doubling.

<https://www.ipcc.ch/site/assets/uploads/sites/2/2019/05/SR15_Chapter1_Low_Res.pdf>
**SPECIAL REPORT: GLOBAL WARMING OF 1.5 ºC  
CH 01  
Framing and Context** The following is on page 66  
Expert judgement based on the available evidence (including model simulations, radiative forcing and climate sensitivity) suggests that if all anthropogenic emissions were reduced to zero immediately, any further warming beyond the 1°C already experienced would likely be less than 0.5°C over the next two to three decades, and also likely less than 0.5°C on a century time scale.

# It seems we already missed the 1.5C target: 3 \* log2(422/280) degree increase, or about 1.78°C

**Tags:** climate-change

> **Comments:**
> **username_1**: Sorry, non-native speaker here, what do you mean with 'allocate [] doublings'? Is it  also possible that you mean not 'doubling' but simply 'increase'? You might get a quicker response if you simplify the language a bit without dumbing it down.  
> **username_0**: @username_1 When CO2 doubles from 280 to 560 we get a 3C increase in global temperature. How much increase do we get when CO2 increases 50% from 280 to 420? I guess 1.5C.  
> **username_2**: [The radiative forcing, and hence the change in temperature, is proportional to the logarithm of the concentration of infrared-absorbing ("greenhouse") gases in the atmosphere, as quantified by Arrhenius in the 19th century. The sensitivity of temperature to atmospheric gasses, most notably CO 2, is often expressed in terms of the change in temperature per doubling of the concentration of the gas.](https://en.wikipedia.org/wiki/Climate_sensitivity#Estimating_climate_sensitivity)  
> **username_3**: It is not as simple as that. I will recommend you [this](http://www.realclimate.org/index.php/data-sources/#Rean_output). In fact, username_3 Archer have a few free courses when you can get additional information, including Python models.  

---

## 2 answers

### Answer by username_4 on 2019-10-29. Score: 1

To get a rough estimate you should be using a logarithmic interpolation rather than a linear interpolation. The response to added $\text{CO}\_2$ would be linear if the atmosphere had barely any $\text{CO}\_2$ in it. Even 280 ppm does not qualify as "very little". "Very little" would mean a handful of parts per million, at most.

The issue is that the atmosphere is rather opaque at those frequencies where $\text{CO}\_2$ absorbs and emits thermal infrared radiation. Instead of a linear response, the response in terms of radiative forcing is roughly logarithmic. This is why climatologists talk about the effect of a doubling the $\text{CO}\_2$ levels. The radiative forcing of an increase from 140 ppm $\text{CO}\_2$ to 280 ppm is more or less the same as the response to an increase from 280 ppm to 560 ppm, or to an increase from 560 ppm to 1120 ppm.

> **Comments:**
> **username_0**: "To get a rough estimate you should be using a logarithmic interpolation rather than a linear interpolation." Yes that is where I thought I was probably wrong. How do we do this?  
> **username_4**: @username_0 - Assuming a 3 °C increase due to a doubling of $\text{CO}_2$ level, the response to a 73% increase (for example) is $3\log_2(1.73) \approx 2.37$, where $\log_2$ is the base 2 logarithm function.  
> **username_0**: How to we account for 2.37C being greater than the reported < 1.5C ?  
> **username_4**: @username_0 - Because the climate has not yet caught up to that increase.  
> **username_0**: I was hoping that I calculated it incorrectly. If my calculations are correct then those of IPCC is wrong or are they only referring to 1.5C as the near term or immediate term feedbacks?  
> **username_4**: Suppose that over the next 24 hours humanity dumps another 140 ppm of $\text{C}_2$ into the atmosphere, and then only emits just enough $\text{CO}_2$ to keep it steady at 560 ppm. The climate will not suddenly change overnight to be 3°C warmer than preindustrial times. It will take several decades for the climate to fully respond to that step change. Climate changes rather slowly. The huge amount of $\text{CO}_2$ humanity has added to the atmosphere over the last 50 years is essentially a step change to which the climate has.not yet fully responded.  
> **username_4**: I was unthinkingly using your 73% value from another question you asked. That would correspond to 484 ppm of $\text{CO}_2$. Using 410 ppm (which is what you used in that other question), the correct math is that one would expect a $3\log_2\left(\frac{410}{280}\right)$ degree increase, or about 1.65°C.  
> **username_0**: That is still more than the claimed: "we are not yet up to 1.5C yet" claim. The transient response is defined by scientists as the temperature response over human time scales of around 70 years, the equilibrium climate sensitivity over centuries, and finally the Earth system sensitivity after multiple millennia.  https://en.wikipedia.org/wiki/Climate_sensitivity#Different_forms_of_climate_sensitivity  
> **username_0**: I just found this. FIGURE 1.5 Warming commitment from past emissions of greenhouse gases and aerosols https://www.ipcc.ch/site/assets/uploads/sites/2/2019/01/figure-5-pdf-e1547458812152.jpg It seems to indicate that IPCC said we already have 1.5C that is on its way. I have to study it more and more to be sure.  
> **username_0**: 3 * log2(396 / 280) = 1.5C. 396 PPM would be 2013 data.  

---

### Answer by username_5 on 2019-12-27. Score: 0

You can use the formula ΔT=λ·α·㏑(C/C₀), (natural logarithm) where C is the later concentration of CO₂ and C₀ the earlier concentration. λ and α is constants. This formula is valid for a solid sphere but since 70 % of the surface of Earth is water, which warms up much deeper and accumulate about 90 % of the accumulated energy, it isn't totally clear how to use the formula correct.

One way could be to use the formula on global land temperatures and transform to global temperature for sea and land afterwards, somehow.

Otherwise, the calculations won't be exact.


---

