Anthropogenic Climate Change: A Scientific Defense of Concern

Matthew PasquiniHarvard UniversityDecember 13, 2014

This paper serves to provide a basic scientific argument to defend an international concern over the effects ofgreenhouse gas emission on global average surface temperature. The issue is seen by many contemporaries aspolitical in nature, and this unfortunate outcome must be transcended to address the issue at hand. Ratherthan political rhetoric, a scientific analysis is the only appropriate tool to accurately determine the likelihoodof human activity as a principal force in global climate change. What follows is an objective presentation ofbasic models which correlate large-scale human carbon emission with a concerning increase in global averagetemperatures.

1 Introduction

Scientists have observed a particularly striking risein the global mean temperature of the Earth’s sur-face, rising over one degree Celsius between duringthe course of the past century. What’s more, this isstrongly correlated with a rise in atmospheric carbondioxide (CO2) concentration by a factor of approxi-mately four-thirds.

[NCDC ]

This correlation alone does not describe a causal re-lationship between carbon dioxide and global meantemperatures. However, simple models enable aninvestigation to show that global average temper-atures are in large part influenced by atmosphericconcentration of greenhouse gases, with a particularemphasis on carbon dioxide.

The model discussed in this paper is limited, however,to global mean temperatures. It does not predict, for

instance, yearly weather cycles, nor does it predictweather by region (as seen in later sections, this isdue to the one-dimensional nature of the model). Itsuse is focused on the firm establishment of climatechange, namely an increase in global average temper-ature on a long time scale (where variabilities averageto this trend).

The model is also limited to long-term temperaturechanges at the surface, and not in the upper atmo-sphere. This is because the model will make the as-sumption of a single thin layer of atmosphere, how-ever the extent of the atmosphere mandates multi-layered models which would more accurately predictupper atmosphere temperatures. Since this paper isfocused on global mean surface temperature analy-sis (the only relevant region in terms of impact onlife), the single-layer model will suffice. [Universityof Texas]

2 Global Energy Balance

Before greenhouse gas models can be prescribed, a ba-sic model that determines average global temperaturemust be established. This can be done through theassumption of global energy balance, that the Earthemits all energy absorbed by the Sun back into space.Satellite measurement has confirmed this is a highlyaccurate assumption [Haar and Suomi ]. Combinedwith an understanding of energy emission from theStephan-Boltzmann law (where emitted radiation isproportional to absolute temperature to the fourthpower), the Earth’s average surface global temper-ature, not considering atmospheric components, is

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determined to be approximately −18 degrees Celsius(See Appendix 7.1 for calculations). [Kuang ]

This is far lower than the true observed value of about14 degrees Celsius [Kivner ]. Therefore, the effect ofthe atmosphere must be taken into account.

3 The Greenhouse Effect

As aforementioned, the model will adopt a singlelayer simplification of the atmosphere to address theissue of surface temperature. Incoming radiationfrom the Sun (after reflection from albedo effects -see Appendix 7.1), in the form of ultraviolet rays,easily penetrates the chemical components of the at-mosphere to reach the surface of the Earth. Whenthe Earth re-radiates this energy at much lower tem-peratures, the energy takes the form of infrared radi-ation, which does not directly penetrate certain com-ponents of the atmosphere. Rather, the energy is re-absorbed and remitted in all directions, with approx-imately half returning to the Earth and half escapingto space.

[Encyclopedia Britannica]

If the fraction of energy captured and reemitted bythe atmosphere (called the absorptivity) is allowed tovary from zero to one, the resultant effect on globalmean surface temperature ranges from no effect toa multiplication by about 1.19, considering the tem-perature’s distance from absolute zero (see Appendix7.2 for a detailed derivation). Below is a plot whichshows this relationship and its effect on global surfacetemperature.

�������������(%)

������������(�)

20 40 60 80 100

-20

-10

10

20

30

Given this plot, an observed surface temperature of14 degrees Celsius would be associated with an ab-sorptivity of about 74 percent. In other words, theatmosphere absorbs approximately three fourths ofthe energy radiated from the Earth, and reradiates ittowards the surface. The question then arises as tothe extent by which human activity has altered thisnumber.

4 The Carbon Impact

One way to determine the relative impact of gases inthe atmosphere is through spectra analysis of radia-tion leaving the Earth, obtainable via satellite. Be-low is Earth’s spectrum as a function of wavenumber(wavelength).

[Kuang ]

The central gap in the spectrum represents the ab-sorption contribution of carbon dioxide, which occu-pies a range of wave number of between around 550to 750 cm−1 [NIST ]. This region, by geometric ob-servation of the spectrum, makes up approximately12 percent of the total area of the spectrum. This is

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a rough approximation, but it will suffice for the pur-pose of this paper, since the goal is determining therelative size of human influence, not the exact impact.

It is now important to observe that carbon dioxidelevels in the atmosphere have risen from 280 to 400parts per million over the past 200 years, measur-able through ice core samples and direct atmosphericmeasurements [University of Michigan]. This is farfaster a change than any natural geologic structurecould produce, and the exponential increase is starklyand visibly corresponding with human industrializa-tion in the 19th and 20th centuries. This is also truefor other, less influential greenhouse gases, as seenbelow.

[University of Michigan]

These trends determine with remarkable confidencethat there is an anthropogenic forcing of greenhousegas accumulation in the atmosphere. That humaninfluence has, conservatively, increased atmosphericcarbon dioxide concentration by a factor of 1.4 meansthat a fraction 1− 1

1.4 ≈ 0.29 of the absorptivity im-pact due to carbon dioxide is anthropogenic. With0.29 × 12% ≈ 3.5%, human influence can be saidto generate about 3.5% of the present atmosphericabsorptivity.

With atmospheric absorptivity found earlier to be74%, without human influence this would be 70.5%.

The greenhouse model (see Appendix 7.2) then pre-dicts a difference in temperature of about 1.96 degreesCelsius. This calculation does not take into accountoptical overlap of carbon dioxide particles (or inter-actions with other gases) and is therefore an upperbound. To achieve a proper lower bound on the iso-lated case of carbon dioxide, the greenhouse model isagain applied but with the assumption that human-generated carbon dioxide is the only greenhouse gaspresent (possible optical overlap in this estimationis offset by the concave-upward curve in the plot oftemperature v. absorptivity). With this assumption,a temperature difference of 1.13 degrees Celsius isdetermined (see Appendix 7.2). An even safer lowerbound could be determined by lowering the influencefrom 3.5% to 2.5% to account for the presence ofother gases affecting the spectrum in question (thereare other dips in the spectrum, at 1000 cm−1 forexample). This produces an impact of 0.80 degreesCelsius.

To summarize, the human carbon impact is deter-mined to occupy an impact roughly between 0.80and 2.00 degrees Celsius. Since the effects of opticaloverlap are likely nontrivial, the value is probablycloser to around one degree. Looking back to Section1 (and converting from Fahrenheit), this rough esti-mate is fairly accurate.

Importantly, this is the impact to date. It does nottake into account further development of carbon diox-ide concentration from current human contributions.For example, if the carbon dioxide concentration wereto increase to 500 parts per million (the earlier figureshows how likely this is), the calculations can be re-peated to show an effect of 1.4 to 2.9 degrees Celsius.

5 Addressing Counter-Theories

Much of the rhetoric dismissing anthropogenic cli-mate change incorporates arguments that there otherfactors scientists haven’t taken into account, or thatrecent cold weather events stand in the face of cli-mate change even occurring.

To address the first issue, perhaps the mostcommonly-used counterargument against the humaninfluence of climate change is that solar activity isdifficult to predict and or determine. While the Sunplays an extremely powerful role in our climate, closeobservation of solar cycles have determined a rela-

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tively stable value of the solar radiation incident uponthe Earth, with periodic cycles determined by solarspot/flare activity.

[Kuang ]

The generally flat trend is in complete disagreementwith global surface temperature observations. Addi-tionally, even if the period cycle were to impact globaltemperatures, the variability of (at most) ±1 W/m

2

translates to a temperature variability of 0.10 degreesCelsius (see Appendix 7.1), a much lower change thanobserved. Therefore, solar activity cannot be thecause of global warming. Anthropogenic greenhousegas forcing remains the proper culprit.

The other common rhetorical argument againstanthropogenic forcing comes with the fact thatseasonal, monthly, and even weekly variability inweather events demonstrates the possibility for ex-cessively cold temperatures, prompting the questionof whether climate change is even occurring. This isflawed in the sense that models such as those pre-sented in this paper determine temperature changeson a global, time-averaged sense. It is impossible forany model to determine all weather-related tempera-ture fluctuations.

Additionally, increase severity of winters does notdiminish the presence of a general warming trend.Warmer global temperatures, especially near thepoles, have the capacity to decrease gradients in tem-peratures with respect to latitude. This reduces driv-ing forces for aloft winds, reducing their resistance tostalling in the jet stream. As a result, “blocking”events in which the jet stream adopts a long-lastingbend promotes the development of more extreme sea-sonal temperature differences, both warm and cold.[Kuang ]

6 Conclusion

The argument for anthropogenic climate changethrough carbon dioxide emission has a clear foun-dation even in basic scientific modeling. Trends inglobal average surface temperature have to date,demonstrated an alarming increase of at least onedegree Celsius so far. Simultaneously, human activ-ity in the past two centuries has increased the carbondioxide content of the atmosphere by a factor of 1.4.Models successfully connect the correlated rise in at-mospheric carbon dioxide concentration to a rise inaverage surface temperature through a mathematicalanalysis of the global energy balance, taking into ac-count the influence of atmospheric absorptivity. Themodel presented in this paper proposes such a rea-sonable and simple approach to find an adjustmentof global mean surface temperature on the order ofone degree Celsius, with a possible range of about 0.8to 2 degrees (additional work could be undergone toaccount more accurately for optical overlap of carbondioxide particles, which would further constrain therange of possible temperature variation). This is instrong agreement with current observation, backingthe well-accepted theory of anthropogenic climateforcing. Additionally, arguments against it such assolar activity and winter severity are easily debunkedthrough additional analysis.

Even with this simple approach, the rise of car-bon dioxide present in the Earth’s atmosphere isnot merely correlated with a coincidental or natu-ral warming of the Earth’s surface, but is, with greatconfidence, the cause thereof. This issue thereforeought no longer remain a political debate, but onewhich the rhetoric of the political stage accepts asscientific coherence and objectivity.

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7 Appendices

7.1 Global energy balance

Define L0 as solar luminosity (energy emitted fromthe sun per unit time), such that the strength of thisradiation spread out across a spherical region in spaceis L0/4πd

2, where d is the distance from the Sun. De-fine α as the albedo (reflectivity) of the Earth, wherea fraction of energy 1−α reaches the surface. There-fore, the energy received by the Earth’s surface perunit time is

Ein =L0

4πd2× πR2 × (1− α) (1)

where R is the radius of the Earth. By the Stefan-Boltzmann Law, the energy released per unit timeper unit surface area by a radiating body is σT 4

where T is absolute temperature and σ is the Stefan-Boltzmann constant. Therefore, the energy emittedby the Earth is

Eout = σT 4 × 4πR2 (2)

Global energy balance requires that Eout = Ein.

L0

4πd2× πR2 × (1− α) = σT 4 × 4πR2 (3)

This solves for T , the global average surface temper-ature.

T =

(L0(1− α)

16πσd2

)1/4

(4)

The quantity L0/4πd2 is often redefined as the Solar

constant S0, a number defining the energy receptionfrom the Sun. Global average surface temperaturecan be rewritten as

T =

(S0(1− α)

4σ

)1/4

(5)

Replacing the appropriate quantities yields

T ≈

((1362 W/m

2)(1− 0.29)

4(5.67× 10−8 W/m2K4)

)1/4

(6)

≈ 255.5 K (7)

≈ −17.5 C (8)

[Kuang ]

7.2 The greenhouse effect

If the incoming solar radiation is assumed to be S0/4,where S0 is the solar constant, an amount (1−α)S0/4will reach the surface without reflecting, and be ab-sorbed. Some amount S will emit from the surfacetowards the atmosphere, which will allow a fraction(1 − ε) into space, where ε is the absorptivity of theatmosphere. The remaining fraction will be reemit-ted equal parts A each into space and towards thesurface.

To maintain radiative balance at the surface (suchthat incoming and outgoing energies equate), the fol-lowing must be true:

S0

4+A =

αS0

4+ S (9)

A similar balance must occur at the top of the atmo-sphere, where the following must hold:

αS0

4+ (1− ε)S +A =

S0

4(10)

This provides a system of equations to solve for Aand S, enabling the determination that:

A =(1− α)εS0

4(2− ε)S =

(1− α)S0

2(2− ε)(11)

Reapplying the Stefan-Boltzmann Law from Ap-pendix 7.1, one can find the surface temperature Ts.

S = σT 4s (12)

=⇒ Ts =

(1

1− ε/2

)1/4((1− α)S0

4σ

)1/4

(13)

In comparison to equation to Eq (5), there exists apre-factor 1/(1− ε/2)1/4. For ε ranging from 0 to 1,this factor ranges from 1 to about 1.19. A plot inSection 3 shows this converted into values in degreesCelsius.

Note that this model is not appropriate for calcu-lations of upper level atmospheric temperatures, asdiscussed in Section 1.

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Later in the paper, the following function is adaptedto calculate global surface temperature as a functionof atmospheric absorptivity ε.

Ts(ε) =

(1

1− ε/2

)1/4

× 255.5 K (14)

The value is then converted to Celsius in the paper.The factor of 255.5 K is taken from the ε = 0 case.The relevant calculations are shown below.

Ts(0.74)− Ts(0.705) ≈ 1.96 K (15)

Ts(0.035)− Ts(0.00) ≈ 1.13 K (16)

Ts(0.025)− Ts(0.00) ≈ 0.80 K (17)

[Kuang ]

8 References

Encyclopedia Britannica, The Greenhouse Effect,2014

Haar, Thomas H. Vander and Verner E. Suomi,Measurements of the Earth’s Radiation Budget fromSatellites During a Five-Year Period, 1971 (Journalof the Atmospheric Sciences)

Kivner, Mark, Global average temperature may hitrecord level in 2010, 2009 (British Broadcasting Com-pany)

Kuang, Zhiming, Notes from The Physics of Climate,2014 (Harvard University)

National Climate Data Center, 2014 (NationalOceanic and Atmospheric Administration)

National Institute of Standards and Technology, Car-bon Dioxide, 2011, (US Department of Commerce)University of Texas at Austin, The Global EnergyBalance

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