Since the current concentrations $N$ of $CO_2$ in the atmosphere is so high, the direct dependence of the surface temperature $T$ on $N$ should be given approximately by $$ T = T_0 + \Delta T {\log{N \over N_0} \over \log 2}\quad\quad\quad\text{(1)} $$ Here $T_0$ is a reference temperature, say the temperature in 1980 and $N_0$ is the corresponding atmospheric concentration of $CO_2$, say 340 ppm at Mauna Loa in 1980. From (1) we see that doubling the CO2 concentration (letting $N = 2N_0$) will increase T by the temperature sensitivity $\Delta T$.
Let's express $\Delta T$: $$ \Delta T = (T - T_0) {\log 2 \over \log (N/N_0)}\quad\quad\quad\text{(2)} $$
In [1]:
%pylab inline
import urllib
Let's fetch the raw data of $CO_2$ measurements at Mauna Loa from the noaa.gov website:
In [2]:
# Only execute this if you want to regenerate the downloaded file
open("data/co2_mm_mlo.txt", "wb").write(urllib.request.urlopen("ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt").read())
Out[2]:
In [3]:
D = loadtxt("data/co2_mm_mlo.txt")
years = D[:, 2]
average = D[:, 3]
interpolated = D[:, 4]
trend = D[:, 5]
As explained in the file co2_mm_mlo.txt, the average column are the raw data of $CO_2$ values averaged over a month, and some months are missing. The trend then subtracts "seasonal cycle" computed over a 7 year window. The missing values in trend are then linearly interpolated. Finally, the average column then contains the trend value plus average seasonal cycle (i.e. average and interpolated contain the same values except for missing months, which are "intelligently" interpolated). We should do this analysis ourselves in the notebook directly from the average data only, but for now let's reuse this analysis.
In [4]:
plot(years, interpolated, "r-", lw=1.5, label="monthly average")
plot(years, trend, "k-", label="trend")
xlabel("Year")
ylabel("$CO_2$ concentration [ppm]")
title("Atmospheric $CO_2$ concentrations at Mauna Loa")
legend(loc="upper left");
Let's get numbers for $N$ (year 2010) and $N_0$ (year 1980) by reading it off the $y$-axis:
In [5]:
idx1980 = sum(years < 1980)
idx2010 = sum(years < 2010)
N0 = trend[idx1980]
N = trend[idx2010]
print("N0 = %.2f ppm (year %.3f)" % (N0, years[idx1980]))
print("N = %.2f ppm (year %.3f)" % (N, years[idx2010]))
Warming from Berkeley Earth in the last 30 years (see a separate notebook for this calculation):
In [6]:
dTdt = 0.24995728742972512 # C / decade
Warming over the past 30 years using satellite measuremenets (see a separate notebook for this calculation):
In [7]:
dTdt = 0.13764588789937693 # C / decade
We'll use the satellite measurements, as arguably they have less systematic errors. Temperature difference:
In [8]:
dT = dTdt * 3 # 3 decades
dT
Out[8]:
From equation (2) we then directly calculate:
In [9]:
from math import log
deltaT = dT * log(2) / log(1.0*N/N0)
print("∆T = ", deltaT, "C")
This assumes that all warming (dT = $T-T_0$) has been caused by $CO_2$ in the past 30 years. If for example only $1/2$ of the warming was caused by $CO_2$, then $\Delta T \approx 1 C$. If on the other hand $CO_2$ stays longer in the atmosphere than other chemicals that might be temporarily cooling the atmosphere down by let's say $1/2$, then $\Delta T \approx 4 C$.