Global Warming and Fourier Transforms

Atmospheric Carbon Dioxide

Situated at 11,135 ft on the north flank of the Mauna Loa volcano on the Big Island of Hawaii, theNational Oceanic and Atmospheric Administration’s Mauna Loa Observatory http://www.mlo.noaa.gov/has been monitoring the level of carbon dioxide in Earth’s atmosphere for over 50 years. The levels of thisgreenhouse gas have been rising steadily during this observation period.

Figure 1: Atmospheric CO2 levels from Ref. [1].

Historical data on the concentration of CO2 in the atmosphere is shown in Fig. 1. After remaining constantat approximately 280 ppm, the concentration has been rising sharply and steadily over the past century.Measurements at Mauna Loa averaged 285 ppm in 2008.

CO2 is not as toxic CO (carbon monoxide) for which the Federal Government has set a safety standard of35 ppm. Concentrations of CO2 above 10, 000 ppm can cause drowsiness, and concentrations above50, 000 ppm will eventually result in death.

The principal reason for concern about the rising CO2 levels in the atmosphere are their effect on theaverage temperature of Earth’s surface and atmosphere. Solar energy incident on Earth’s is partiallyreflected back into space as lower wavelength infrared radiation, see Fig. 2. CO2 in the atmosphere tendsto trap this radiation and is an important factor in the phenomenon of global warming. Global warminghas important consequences for the biosphere and human society. Every concerned scientist should readthe reports of the Intergovernmental Panel on Climate Change[3].

The Fourier Transform

The concept of Fourier transformation was introduced by Joseph Fourier in his study On the Propagationof Heat in Solid Bodies published in 1807. He found that the heat conduction equation

∂

∂tT (x, t) = κ

∂2

∂x2T (x, t) , (1)

1

Figure 2: Solar energy balance from Ref. [3].

where T is the temperature distribution in a linear conductor of length L with its ends held at fixedtemperatures, could be solved by expanding T (x, t) as a sum of trigonometric functions

T (x, t) = c0 + c1x+∞∑n=0

ane−κλ2

nt sin (λnx) , (2)

where λn = nπ/L, and the Fourier coefficients are determined by the Fourier integral transform

an =2L

∫ L

0

dx [T (x, 0)− c0 − c1x] sin(λnx) (3)

from the initial temperature distribution.

Discrete Fourier Transform

Suppose we measure the temperature inside the rod at N equally spaced points xj = jL/Nj = 0, 1, 2 . . . N−1. These N values can be expressed exactly as a finite Fourier sum

T (xj)− T (0) = Tj =N−1∑k=1

Fk sin(πkj

N

), (4)

where the discrete Fourier coefficients are given by

Fk =2N

N−1∑j=1

Tj sin(πkj

N

). (5)

This discrete sine transform can be written in matrix form

~T = S~F , ~F =2N

S~T , (6)

where the matrix S has elements Sjk = sin(πjk/N).

Computing such a discrete Fourier transform involves O(N2) operations, because there are N − 1coefficients each represented as a sum of N − 1 terms. The FFT (Fast Fourier Transform) algorithmcomputes the transform in O(N log2N) operations. This is a significant speedup for large vectors:

2

N Discrete Fourier Transform Fast Fourier Transform103 1.0 sec 0.01 sec106 106 sec = 12 days 20 sec109 1012 sec = 32, 000 years 3.0× 106 sec = 8.3 hours

Fourier integral formulas

A piece-wise continuous square-integrable complex function f(t) in [−∞ < t < +∞] can be represented asa Fourier integral over an angular frequency domain [−∞ < ω < +∞]:

f(t) =1√2π

∫ ∞−∞

g(ω)eiωtdω , (7)

where the Fourier transform

g(ω) =1√2π

∫ ∞−∞

f(t)e−iωtdt . (8)

Parseval’s Theorem for the total power of g(t) states

Total Power =∫ ∞−∞|f(t)|2dt =

12π

∫ ∞−∞|g(ω)|2dω . (9)

The term power can be understood by considering the example f(t) = I(t) where I(t) is the current in anelectrical circuit with resistance R: the instantaneous power dissipated is I2R ∼ |I(t)|2, and the “totalpower” is therefore the energy dissipated. A very useful quantity is the one-sided power spectral densitywhich measures the total power per unit frequency at |ω|:

P (ω) ≡ |g(ω)|2 + |g(−ω)|2 , 0 ≤ ω <∞ . (10)

C++ Program to Analyze CO2 Data

Program 1: http://www.physics.buffalo.edu/phy410-505/topic1/co2.cpp

#include <cmath>#include <cstdlib>#include <fstream>#include <iostream>#include <string>#include <vector>using namespace std;

vector<double> sine_transform(const vector<double>& data

) {int N = data.size();vector<double> sine_transform(N);const double pi = 4 * atan(1.0);for (int k = 1; k < N; k++)for (int j = 1; j < N; j++)sine_transform[k] += data[j] * sin(pi * j * k / double(N));

3

return sine_transform;}

int main() {

string data_file_name("maunaloa.co2");ifstream data_file(data_file_name.c_str());if (data_file.fail()) {cerr << "cannot open " << data_file_name << endl;exit(EXIT_FAILURE);

}

string word;dodata_file >> word;while (word != "1958");

vector<double> ppm_data;for (int year = 1958; year <= 2008; ++year) {double ppm;for (int month = 0; month < 12; month++) {data_file >> ppm;if (ppm > 0)ppm_data.push_back(ppm);

}data_file >> ppm >> word;

}cout << " read " << ppm_data.size()

<< " values from " << data_file_name << endl;

vector<double> sft_data = sine_transform(ppm_data);

string plot_file_name("plot.data");ofstream plot_file(plot_file_name.c_str());for (int k = 1; k < sft_data.size(); k++)plot_file << k << ’\t’ << sft_data[k] << ’\n’;

plot_file.close();cout << " wrote sine transform in " << plot_file_name << endl;

}

Homework Problem

The Mauna Loa CO2 concentration increases with every passing year in addition to oscillating with theseasons. To Fourier analyze the seasonal oscillation, the linearly increasing trend needs to be subtractedfrom the data. Fit the concentration data to a quadratic function of time f(t) = a0 + a1t+ a2t

2 using ageneral linear fit routine. Is the linear increase accelerating or decelerating? If the current trends continue,when will the concentration reach toxic levels? Next, subtract the fit function f(t) from the data. Plot the

4

Figure 3: Power spectrum of Mauna Loa CO2 concentration.

subtracted data and its power spectrum. Compare with the original plots and interpret your results.

References

[1] J.L. Sarmiento and N. Gruber, “Sinks for Anthropogenic Carbon”, Physics Today, 55, 30 (August2002), http://dx.doi.org/10.1063/1.1510279,http://www.physics.buffalo.edu/phy410-505/topic1/carbon_sinks.pdf.

[2] R.F. Keeling, S.C. Piper, A.F. Bollenbacher and J.S. Walker, “Atmospheric Carbon Dioxide Recordfrom Mauna Loa”, CDIAC http://cdiac.ornl.gov/trends/co2/sio-mlo.html

[3] Report of the Intergovernmental Panel on Climate Change (2007) http://www.ipcc.ch/.

[4] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, “Numerical Recipes in C” (CambridgeUniversity Press 1992), §12.2 Fast Fourier Transform (FFt),http://www.nrbook.com/a/bookcpdf/c12-2.pdf, §12.4 Fast Sine and Cosine Transforms,http://www.nrbook.com/a/bookcpdf/c12-4.pdf.

5