RESOURCE LETTER

Resource Letters are guides for college and university physicists, astronomers, and other scientists to literature, websites, and other teaching aids.

Each Resource Letter focuses on a particular topic and is intended to help teachers improve course content in a specific field of physics or to

introduce nonspecialists to this field. The Resource Letters Editorial Board meets annually to choose topics for which Resource Letters will be

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general interest to persons seeking to become informed in the field, the letter I to indicate intermediate level or somewhat specialized material, or the

letter A to indicate advanced or specialized material. No Resource Letter is meant to be exhaustive and complete; in time there may be more than one

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www/Readers/resLetters.html>. Suggestions for future Resource Letters, including those of high pedagogical value, are welcome and should be sent

to Professor Mario Belloni, Editor, AJP Resource Letters, Davidson College, Department of Physics, Box 6910, Davidson, NC 28035; e-mail:

mabelloni@davidson.edu.

Resource Letter GECC-2: The Greenhouse Effect and Climate Change:The Intensified Greenhouse Effect

Stephen E. SchwartzEnvironmental and Climate Sciences, Brookhaven National Laboratory, Upton, New York 11973

(Received 15 June 2018; accepted 17 June 2018)

Human activities over the past 200 years have increased Earth’s greenhouse effect by about 1%

relative to the radiative fluxes that drive the climate system. This Resource Letter introduces the

physics of this intensified greenhouse effect and of the processes that govern resultant change in

climate and provides resources for further study. Context for this examination is provided in a

companion Resource Letter (GECC-1) that examined the radiative fluxes that comprise Earth’s

climate system and the present-day greenhouse effect. The increase in global temperature and other

changes in climate resulting from the intensified greenhouse effect are of great societal concern.

Developing prognostic capability to determine these responses to the small perturbations that constitute

the intensified greenhouse effect to an accuracy that would be useful to inform policymaking is the

major challenge facing climate scientists today. VC 2018 American Association of Physics Teachers.

https://doi.org/10.1119/1.5045577

I. INTRODUCTION

As described in the companion Resource Letter (GECC-1), the so-called “greenhouse effect” results in an increase inglobal mean surface temperature (GMST) that is about 32 Kgreater than what it would otherwise be for the same plane-tary absorption of solar radiation and is thus a major featureof Earth’s present climate. The examination in GECC-1 ofthe magnitudes of the several energy fluxes that characterizeEarth’s climate system, the physical processes that give riseto the greenhouse effect, and the magnitude of the green-house effect provides context and sets the scene for theexamination here of the increases of atmospheric amounts ofgreenhouse gases (GHGs) and other changes in atmosphericcomposition due to human activities over the Anthropoceneepoch,1–3 the period of time over Earth’s biogeochemistryand climate have been appreciably affected by human activi-ties, roughly the past 200 years, and the consequences ofprior and prospective changes in climate. The emphasis herewill be on change in GMST. It is commonly assumed, andgenerally found in studies with large-scale climate models,that changes in other climate properties, such as precipita-tion, will be linear in the change in GMST, but with recogni-tion that the changes in climate variables will not bespatially uniform.

Sources and References. As indicated in Resource LetterGECC-1, a broad and detailed examination of present under-standing of the climate system and climate change is given

in the several Assessment Reports of Working Group I of theIntergovernmental Panel on Climate Change (IPCC), espe-cially the two most recent reports.4,5 These reports, whichare freely downloadable on the web, are essential referencesfor any student of climate and climate change. ResourceLetter GECC-1 called attention also to several texts thatmight serve as an introduction to the climate and to the phys-ics of climate. A hypertext history of the development ofunderstanding of the greenhouse effect and climate changemore generally by Weart,6 which supplements his very read-able book, may be accessed at http://www.aip.org/history/climate/index.htm. The monograph by Br€onnimann7 dealswith many aspects of natural and anthropogenic climatechange over the past 500 years. Citations to the primary andsecondary literature given in this Resource Letter areintended to be of historical and pedagogical value, as well asto lead the reader to the research literature without attempt-ing to be exhaustive.

Geophysical data. Data sets particularly pertinent to cli-mate change over the Anthropocene in addition to thosegiven in Resource Letter GECC-1 are time series for compo-nents of global mean radiative forcing and for componentsof the global atmospheric CO2 budget; see also citations todata sources given in captions to Fig. 1 and Table II.

Organization of this Resource Letter. Section II quantifiesthe increases in amounts of GHGs over the Anthropocene,introduces the radiative forcing concept, and quantifies forc-ings by incremental GHGs and other forcing agents over the

645 Am. J. Phys. 86 (9), September 2018 http://aapt.org/ajp VC 2018 American Association of Physics Teachers 645

BNL-205797-2018-JAAM

Anthropocene. Section III focuses more explicitly on forc-ings by incremental GHGs and on the atmospheric residencetimes of the key GHGs. Section IV examines climate systemresponse to forcings, introduces the climate sensitivity con-cept, and examines estimates of Earth’s climate sensitivityand present uncertainty in these estimates. Section V exam-ines the expected change in global temperature that would beexpected based on estimates of forcings over theAnthropocene together with estimates of Earth’s climate sen-sitivity, again with associated uncertainties and examinesimplications for prospective future change in global tempera-ture. A summary is presented in Sec. VI.

Supplementary Notes (SN) in the online version of thisResource Letter provide detail or lend perspective, as follows.

SN1. Calculation of radiative forcing by an incrementalgreenhouse gas.SN2. Can forcings from increases in GHGs be measured?SN3. The budget and adjustment time of incremental atmo-spheric CO2.SN4. Treating global temperature response as linear in theperturbation.

1. “The Anthropocene,” P. J. Crutzen and E. F. Stoermer,IGPP Newsletter No. 41, pp. 16–18 (2000). http://www.igbp.net/download/18.316f18321323470177580001401/1376383088452/NL41.pdf. Historical. (E)

2. “Geology of mankind,” P. J. Crutzen, Nature 415, 23–23(2002). https://www.nature.com/articles/415023a. Historicaland influential commentary. (E)

3. “Theme Issue ‘The Anthropocene: A new epoch of geo-logical time?’,” compiled and edited by M. Williams, J.Zalasiewicz, A. Haywood, and M. Ellis, Philos. Trans. R.Soc. A, Vol. 369, Issue 1938, 2011. http://rsta.royalsocie-typublishing.org/content/369/1938. Thirteen articlesexamining aspects of the Anthropocene, from geologicalto societal. (E)–(I).

4. Climate Change 2013: The Physical Science Basis

(AR5), edited by T. F. Stocker et al. (Cambridge U.P.,Cambridge, United Kingdom and New York, NY, 2013).http://www.climate2013.org/images/report/WG1AR5_ALL_FINAL.pdf. Intergovernmental Panel on Climate Change.Essential reference. (E)–(I).

5. Climate Change 2007: The Physical Science Basis

(AR4), edited by S. Solomon et al. (Cambridge U.P.,Cambridge, United Kingdom and New York, NY, 2007).http://www.ipcc.ch/publications_and_data/ar4/wg1/en/contents.html. Intergovernmental Panel on ClimateChange. Essential reference. (E)–(I).

6. The Discovery of Global Warming: Revised and

Expanded Edition, Spencer R. Weart (Harvard U.P.,2008). Good introduction to climate, greenhouse effect,and intensified greenhouse effect. (E)

7. Climatic Changes since 1700, S. Br€onnimann (Springer,Cham, 2015). DOI 10.1007/978-3-319-19042-6. Discussesmany elements of climate and climate change on a varietyof temporal and geographical scales. (E)–(A).

8. Supporting Information to IPCC AR5 Chapter 8, AnnexII: Climate System Scenario Tables, Table AII.1.2,“Historical effective radiative forcing (ERF) (W�m�2),including land use change (LUC)” Published 31 January2014; corrected xls-sheets as of 20 August 2014. http://www.climatechange2013.org/images//report/WG1AR5_AIISM_Datafiles.xlsx.

9. “Global Carbon Budget 2015,” C. Le Qu�er�e et al., EarthSystem Science Data, 7, 349–396 (2015). http://cdiac.ess-dive.lbl.gov/ftp/Global_Carbon_Project/Global_Carbon_Budget_2015_v1.1.xlsx

II. RADIATIVE FORCING OF CLIMATE CHANGE

The anthropogenic influences on the planetary radiationbudget and responses of the climate system to perturbations

Fig. 1. Atmospheric mixing ratios (left axes) and resulting global-mean perturbations in radiative budget (radiative forcings, right axes) of carbon dioxide,

methane, and nitrous oxide over (a) the last glacial-interglacial cycle, 150,000 years; (b) the Holocene, 10,000 years; (c) the Anthropocene, 250 years, and (d)

contemporaneous measurements at Mauna Loa, Hawaii. Measurements from Antarctic ice cores in (b) and (c) are shown as symbols with different levels of

gray (color online) for different studies; contemporaneous atmospheric measurements are shown by thin curves in (d) and corresponding gray curves in other

panels (color online). (a) CO2, Ref. 12; CH4, Ref. 13; (b) and (c) modified from the Fourth IPCC Assessment Report (Ref. 5, Sec. 6.4.1.1); (d) monthly mean

data from NOAA (ftp://aftp.cmdl.noaa.gov/data/trace_gases/) and, for CO2, Scripps Institution of Oceanography (http://scrippsco2.ucsd.edu/data/atmospheric_

co2/primary_mlo_co2_record), updated from Ref. 14.

646 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 646

are the key questions that face the climate change researchcommunity today. Well established by measurements, atmo-spheric mixing ratios of several long-lived gases that contrib-ute to Earth’s greenhouse effect have increased substantiallyover the past 250 years because of human activities [seeFig.1]. Mixing ratios prior to contemporaneous measure-ments are from glacial ice cores, mainly in Antarctica, forwhich the record extends to 800,000 years before the present.The dataset initiated in 1958 by Charles David Keeling10,11

at Mauna Loa, Hawaii, isolated from local sources and sinksand thus representative of the Northern Hemisphere, revealsthe annual cycle of draw down and release of CO2 by terres-trial vegetation. This dataset has become an icon of theanthropogenic influence on GHG concentrations. Theincrease in abundance of carbon dioxide and methane overthe Anthropocene is comparable to the increase between thelast glacial maximum and the Holocene, the current temper-ate epoch of Earth’s geological history. Shown on the right-hand axes of the several panels in Fig. 1 are the perturbationsin the longwave radiation budget of the planet that wouldresult from the increases in the mixing ratios of the severalgases. This hypothetical increase in the net rate of uptake ofenergy by the climate system (or more generally, any suchchange in Earth’s energy budget that is imposed from outsidethe climate system) is denoted a forcing; like the energyfluxes themselves, forcings are commonly given normalizedto the area of the planet, with unit W�m�2. The sum of theseveral forcings during the Anthropocene due to the increase

in abundance of GHGs leads to expectation of a resultantincrease in the greenhouse effect and consequent increase inEarth’s surface temperature. The increases in atmosphericabundances of the so-called long-lived GHGs, most impor-tantly carbon dioxide (CO2), but also including methane(CH4), nitrous oxide (N2O), and chlorofluorocarbons(CCl2F2, CCl3F) are at the core of the present concern overanthropogenic global warming, or more broadly over anthro-pogenic climate change.

The magnitude of forcing by the several long lived GHGs,about 3 W�m�2 at present relative to the preindustrial atmo-sphere, is some two orders of magnitude less than the globaland annual mean outgoing longwave flux at the top of theatmosphere (TOA) or than the corresponding averagedownwelling or upwelling longwave fluxes at the surface(Resource Letter GECC-1, Fig. 2). This disparity serves tojustify treating this forcing as a perturbation on the climatesystem. Moreover, the spatial and temporal variations in theseveral fluxes inherent in the climate system (ResourceLetter GECC-1, Fig. 3) greatly exceed the magnitude of theperturbation due to increased amounts of atmosphericGHGs, challenging the capability to measure the radiativeeffects of the incremental GHGs (SN2). Consequently, theseradiative effects are not determined by measurement but arecalculated, based on the absorption spectra of the severalgases and the atmospheric state.

Over the Anthropocene, industrial activities have alsoresulted in increases in the amounts of aerosols in the

Fig. 2. Radiative forcings, relative to 1750, by key climate influencing agents over the Anthropocene epoch, left, and for 2011, numerical values and bars at

right. Bar denoting forcing by tropospheric aerosols represents the sum of direct aerosol interaction with radiation (Aer-Rad Int.) and indirect interaction

through aerosol cloud interactions (Aer-Cld Int.). Uncertainties on bars represent central 90% range of likelihood distribution. Large negative spikes denote

forcing by aerosols, mainly in the stratosphere, resulting from major volcanic eruptions; magnitudes of early volcanos are truncated. Modified from IPCC (Ref.

4), Fig. 8.18 by addition of numerical values from Table 8.6 and of names of major volcanos.

Fig. 3. Global average radiative forcing by long-lived greenhouse gases as a function of their mixing ratios (parts per million, ppm; billion, ppb; trillion, ppt) shown

on abscissa beginning at preindustrial values and extending approximately to present values. Curves (red, online) forcing calculated by approximate formulas (Refs.

18 and 19). Lines (blue, online) denote linear dependence of forcing on mixing ratio. Note differences in horizontal and vertical scales for the several gases.

647 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 647

troposphere. Aerosol particles scatter shortwave radiation andincrease the reflectivity and persistence of clouds; increases inaerosol loadings would be expected to reduce the absorption ofshortwave radiation incident on the planet, thereby changingEarth’s radiation budget (i.e., exerting a forcing) in a directionopposite to the increase in net energy due to increased GHGs.The magnitudes of the radiative perturbations due to anthropo-genic aerosols are comparable to those of the GHGs, but theperturbations are much less well understood and quantified;absorption of shortwave radiation by soot particles lends fur-ther complexity to the aerosol forcing. Incremental tropo-spheric aerosols are due largely to fossil fuel combustionthrough direct emission and atmospheric reactions involvingsulfur and nitrogen oxides emitted as combustion byproducts.A major difference between the influences of incrementalGHGs and incremental aerosols arises from the vast differencein atmospheric residence times of the GHGs (decades to centu-ries) and the tropospheric aerosols (about 10 days, throughremoval mainly by precipitation), the implications of whichare examined in Sec. V. Quantitative understanding of the cli-mate system response to perturbations over the Anthropoceneand to responses to future changes in atmospheric compositionmust take all of these perturbations into account. As the aerosolinfluences are not well understood, they are the subject ofmuch current research.15 Present best estimates by the IPCC4

of global-mean forcings of Earth’s radiation budget over theAnthropocene epoch are shown in Fig. 2. The sign conventionis that forcings, which increase the net energy of the planet, aredenoted as positive and those which decrease the net energyare negative; thus, forcings by incremental GHGs are positiveand forcings by incremental aerosols are in the aggregate nega-tive. As discussed in Sec. IV and SN4, change in GMST inresponse to forcings can be considered in good approximationto be linear, at least to first order; that is for a forcing of a giventype the response is proportional to the strength of the forcing,and forcings of different types are algebraically additive. Thus,the negative forcing by anthropogenic aerosols must be consid-ered as offsetting some fraction of the positive forcing byGHGs.

In addition to the gradual, longer-term perturbations dueto the build up of atmospheric GHGs and aerosols over theAnthropocene, the total forcing has been punctuated by inter-mittent large negative forcings arising from major volcaniceruptions. These eruptions inject sulfur dioxide into thestratosphere; sulfuric acid aerosol formed from this sulfurdioxide scatters incident shortwave radiation, thus increasingthe planetary albedo. The forcing from a given volcaniceruption decreases gradually over a several-year period asthe aerosol is removed from the stratosphere as stratosphericair is exchanged with that of the troposphere; once in the tro-posphere the aerosol is rapidly (weeks) removed from theatmosphere, mainly by precipitation.

Shown at the right of Fig. 2 are the best estimates of thevalues of present (2011) forcings by the several forcingagents (bars) and of the associated uncertainties (I-beamsdenoting central 90% of likelihood distribution) given by the2013 IPCC assessment.4 The relatively small uncertaintiesassociated with forcings by the incremental GHGs contrastto the quite large uncertainties associated with the incremen-tal tropospheric aerosols reflected in the large uncertaintyrange given in the Assessment [�0.09 to �1.88 W�m�2].Because of the additivity of forcings, the best estimate oftotal forcing carries with it the uncertainty associated withthe aerosol forcing. If the aerosol forcing is at the low-

magnitude end of the indicated range, then the aerosol is off-setting only a small fraction of the forcing by GHGs, and thetotal forcing is at the high end of the indicated range,3.33 W�m�2. If, however, the aerosol forcing is at the high-magnitude end of the uncertainty range, the offset of GHGforcing by aerosol forcing is quite substantial, with the totalforcing 1.13 W�m�2. The resultant uncertainty in total forc-ing is thus a factor of 3. As discussed in Sec. IV this uncer-tainty in total forcing over the Anthropocene epoch hasmajor implications with respect to interpretation of climatechange over this period.

10. “Atmospheric carbon dioxide variations at Mauna LoaObservatory, Hawaii,” C. D. Keeling, R. B. Bacastow,A. E. Bainbridge, C. A. Ekdahl, Jr., P. R. Guenther, L. S.Waterman, and J. F. S. Chin, Tellus 28, 538–551 (1976).http://www.tandfonline.com/doi/abs/10.3402/tellusa.v28i6.11322. Landmark paper by the pioneer in the field. (E)

11. “Exchanges of atmospheric CO2 and 13CO2 with the ter-restrial biosphere and oceans from 1978 to 2000. I.Global aspects,” C. D. Keeling, S. C. Piper, R. B.Bacastow, M. Wahlen, T. P. Whorf, M. Heimann, andH. A. Meijer, SIO Reference Series, No. 01–06, ScrippsInstitution of Oceanography, San Diego (2001). https://cloudfront.escholarship.org/dist/prd/content/qt09v319r9/qt09v319r9.pdf. Update of record and methods and inter-pretation. (E).

12. “High-resolution carbon dioxide concentration record,650 000–800 000 years before present,” D. L€uthi et al.,Nature 453, 379–382 (2008). (E)

13. “Orbital and millennial-scale features of atmosphericCH4 over the past 800,000 years,” L. Loulergue et al.,Nature 453, 383–386 (2008). (E)

14. “Interannual extremes in the rate of rise of atmosphericcarbon dioxide since 1980,” C. D. Keeling, T. P. Whorf,M. Wahlen, and J. V. D. Plicht, Nature 375, 666–670(1995). https://www.nature.com/articles/375666a0. Akey research paper by one of the most important figuresin the field. (I)

15. “Aerosol properties and processes: A path from field andlaboratory measurements to global climate models,” S. J.Ghan and S. E. Schwartz, Bull. Am. Meteorol. Soc. 88,1059–1083 (2007). http://journals.ametsoc.org/doi/abs/10.1175/BAMS-88-7-1059. Review of processes con-trolling amount and properties of atmospheric aerosols.(E)–(I)

III. THE INTENSIFIED GREENHOUSE EFFECT

The foregoing overview of the climate system, with empha-sis on the processes that control Earth’s radiation budget, per-mits examination of the anthropogenic perturbations to thisbudget. The important drivers of this change are increases inatmospheric infrared-active gases over the Anthropocene,shown in Fig. 1, and increases in tropospheric aerosols. Thissection examines the changes in amounts of greenhouse gasesand the resultant forcings. Determination of the changes inEarth’s radiation budget due to perturbations in compositionrequires knowledge of the increments in atmospheric abun-dances of the pertinent substances and evaluation of the result-ing perturbations in radiation. The forcings by incrementalgreenhouse gases are, to a good approximation, linear in theamount of incremental gas. It is thus possible to separate thecalculation of the forcing into two components: (1)

648 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 648

determination of the anthropogenic perturbation to the amountof greenhouse gas, and (2) determination of the change inTOA flux per increment of the gas. This separation simplifiesthe problem and permits attribution of the sources of uncer-tainty to one or the other component.

A. Anthropogenic increments in greenhouse gases

As shown in Fig. 1, the amounts of carbon dioxide, meth-ane, and nitrous oxide, which had been relatively constantfor several thousand years, have increased substantially andsystematically over the past two centuries, a consequence ofhuman activities. These increments are firmly established bymeasurements in ice cores and, for more recent times, directatmospheric measurements. The increase in the amount ofCO2 in the first half of the 19th century was due mainly todeforestation;16 subsequently this increase has been increas-ingly dominated by emission from fossil fuel combustion,which now accounts for almost 90% of total CO2 emissions.The increases in methane and nitrous oxide are due to a mixmainly of agricultural activities and energy production,Table I. Additionally, chlorofluorocarbons CCl2F2 andCCl3F, which are also strongly infrared-active, have beenintroduced into the atmosphere through their use as refriger-ants and propellants. Although production of these gases hasbeen greatly curtailed because of their role in depletion ofstratospheric ozone, these gases remain present in the atmo-sphere17. These changes in atmospheric composition haveresulted in a change Earth’s radiation budget, mainly in thelongwave, and thus a radiative forcing of climate change.

B. Radiative forcing by incremental greenhouse gases

The radiative forcing due to an increase in the amount of agreenhouse gas in the atmosphere is the net decrease in

outgoing longwave radiation at the TOA that would resultfrom such an increase prior to any increase in Earth’s surfacetemperature that would counteract the radiative imbalanceimposed by this increase. This forcing is determined by radi-ation transfer calculations that rely on the spectroscopicproperties of the several gases. Such radiation transfer calcu-lations are carried out with global-scale models that haverealistic distributions of temperature, clouds, and watervapor. Integration over space and over an annual cycle (orperhaps several years to average out fluctuations) yields theforcing for a given increment of greenhouse gas or gases.The example calculation of radiative forcing in SN1 explic-itly illustrates the dependence of this forcing on the verticaltemperature structure of the atmosphere.

Calculations such as those in SN1 but carried out globallyand over annual and diurnal cycles, have led to developmentof simplified expressions for the global mean forcings of theseveral long-lived GHGs given in Table II. The dependenceof CO2 forcing on mixing ratio shown in the table as loga-rithmic is sublinear in mixing ratio (slope decreases withincreasing value of argument), as the increase in absorptionis due only to increased absorption in the wings of the lines,the line centers being saturated. For CH4 and N2O the linesare not quite so saturated, resulting in an increase in forcingwith increasing mixing ratio that is less sublinear than that ofCO2 and which is commonly represented as a square-rootdependence. For the chlorofluorocarbons, CCl2F2 and CCl3F,which are entirely anthropogenic and for which the base-case mixing ratio is zero, the forcing is linear in mixing ratioand is much stronger per incremental mixing ratio than forCO2. Although commonly given expressions for forcing bythe incremental GHGs are sublinear, the forcings to date, rel-ative to the preindustrial climate, are approximately linear,Fig. 3, although sublinear expressions would be needed forwider ranges of mixing ratios. The linearized expressionsgiven in Table II explicitly show the much stronger forcingper increment of mixing ratio for chlorofluorocarbons, fol-lowed by N2O and CH4 than for CO2. It is this variation inforcing per incremental amount of gas in atmosphere, whichgets entirely obscured by the several different functionalforms, that is much more the take-home message than anynonlinear dependence on mixing ratio.

Because of the importance of forcing by GHGs as a driverof climate change, and because forcing cannot be directlymeasured (SN2) but can be determined only by radiativetransfer calculations, such calculations assume considerablesignificance. A measure of the accuracy of such calculationscomes from comparisons of forcings as calculated by multi-ple models. The global mean forcing that would result fromdoubling the atmospheric mixing ratio of CO2, as calculated

Table I. Fractional contributions by source category, per cent, to total cur-

rent anthropogenic emissions of CO2, CH2, and N2O. Based on emissions

estimates provided by IPCC (Ref. 4), Chapter 6.

Gas

Anthropogenic source CO2 CH4 N2O

Fossil-fuel related & cement production 87 29 3

Net deforestation & Biomass burn 13 11 12

Total Agriculture & Fertilizer … 38 81

Animal husbandry … 27 …

Rice cultivation … 11 …

Landfills … 23 …

Human excreta … … 3

Table II. Mixing ratios, forcings, and adjustment times of long-lived greenhouse gases. Mixing ratio x is in parts per billion, ppb; subscript 0 denotes preindus-

trial. Expressions for forcings from Ref. 18. Linear approximations are shown in Fig. 3. Mixing ratios and forcings are from NOAA annual greenhouse gas

index http://www.esrl.noaa.gov/gmd/aggi/aggi.html; accessed 2017-11-17. Adjustment times from IPCC (Ref. 4; Sec. 8.3) except for CO2 (Refs. 20, 21; SN3).

Trace gas

Preindustrial

mixing ratio,

x0, ppb

2016 mixing

ratio, x, ppb

Expression for

forcing, W�m �2

2016 forcing

W�m �2

Slope of linear

fit, W�m �2�ppb�1

Adjustment

time yr

CO2 278� 103 403� 103 5:35ðlnx� lnx0Þ 1.99 0.0000159 40 – 50

CH4 700 1843 0:036ðffiffiffixp� ffiffiffiffiffi

x0p Þ 0.51 0.00051 12

N2O 270 329 0:12ðffiffiffixp� ffiffiffiffiffi

x0p Þ 0.19 0.0035 120

CCl2F2 0 0.512 0.33 ðx� x0Þ 0.16 0.33 100

CCl3F 0 0.230 0.25 ðx� x0Þ 0.06 0.25 45

649 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 649

for a cloud-free atmosphere by several radiation transfermodels and by the radiation codes of several general circula-tion climate models, was found to exhibit quite small spread(standard deviation 0.07 W�m�2 or 1.6%). This small spreadis indicative of the confidence that can be placed in such cal-culations.22 However, as the comparison was conducted for acloud-free planet in order to avoid complications arisingfrom differing treatments of clouds in climate models, theresulting forcing, about 4.5 W�m�2, substantially exceeds the3.7 W�m�2 for forcing by doubled CO2 obtained with thewidely used expression in Table II. Moreover, it has becomeappreciated in the past several years that the introduction ofa greenhouse gas into the atmosphere would likely inducechanges in atmospheric thermal structure and in turn clouds,thereby modifying the radiative impact of the incrementalgreenhouse gas from what it would be with static atmo-spheric structure. As this adjustment occurs rapidly (days toweeks), well prior to appreciable response of global surfacetemperature to the forcing,23,24 the resultant radiativechanges are more usefully viewed as altering the forcingrather than as a response of the climate system to the forcing.The resulting forcing has been denoted the “adjusted for-cing.” The inter-model spread in adjusted forcing by doubledCO2 is much greater than that obtained for the pure radiativeforcing in the absence of clouds and without accounting foradjustments. For the 23 climate models that participated inthe 2013 IPCC model comparisons, the adjusted forcingexhibited a mean of 3.44 W�m�2, and 90% uncertainty range[6 0.84 W�m�2], equivalent to [6 24%]. This spread in esti-mated forcing is close to the IPCC4 estimate of 90% uncer-tainty range in forcing by long-lived GHGs [6 20%]. Ofcourse, the question always remains whether spread in modelcalculations is an accurate estimate of uncertainty, but it is insome sense a lower bound to the uncertainty that can beascribed to present estimates.

One further greenhouse gas that should be mentioned istropospheric ozone, which is produced by atmosphericchemical reactions involving nitrogen oxides emitted asbyproducts of fossil fuel combustion and natural and anthro-pogenic hydrocarbons. Concentrations of ozone haveincreased substantially over the past 250 years. As produc-tion is localized to areas of high emission density and exhib-its a rather strong seasonal cycle, and as excess atmosphericozone is rather short lived (months), the gas is rather nonuni-formly distributed spatially and temporally. The best esti-mate of the greenhouse forcing by incremental troposphericozone is about 0.4 6 0.2 W�m�2; a forcing of this magnitude,although not negligible as an agent of climate change, issmall relative to forcing by the incremental long-livedGHGs. The large relative uncertainty, [6 50%] (central 90%of the likelihood distribution), is due much more to uncer-tainty in the amount and distribution of the increase in tropo-spheric ozone over the Anthropocene than to uncertaintiesassociated with radiative transfer.

Two additional points should be noted regarding forcings.First, as indicated in Sec. II and examined further in Sec. IV,the forcing by increases in GHGs over the Anthropocenerepresents the amount by which the longwave energy leavingthe planet would have decreased, relative to the preindustrialsteady state, in the absence of any increase in global temper-ature and resultant increase in longwave emission (and/ordecrease in shortwave absorption) that would offset theimposed imbalance. As discussed in SN1 of Resource LetterGECC-1, a forcing of 3 W�m�2, if not compensated by

restoration of the planetary energy balance resulting fromincrease in surface temperature, would result in a planetarywarming rate much greater than has been experienced overthe Anthropocene. Thus, the change in net irradiance at thetop of the atmosphere (TOA) in response to this applied forc-ing (i.e., the planetary energy imbalance) must be well lessthan the applied forcing, and this would apply in general forthe planetary response to any applied forcing.

A second important point about the forcing by increases inGHGs over the Anthropocene is that this forcing represents aperturbation of about 1% of the total global and annual meanoutgoing longwave radiation at the TOA. The requirement isthus to determine the consequences of a 1% change in aquantity that is highly variable spatially and temporally tosome desired accuracy, say 25%. This is a key reason whyquantifying the effects of the increases in GHGs on Earth’sclimate has been and continues to be such a challenge.

In summary, current estimates of radiative forcing byincremental GHGs, at present and as a function of time overthe Anthropocene, rest on radiative transfer calculations,averaged globally and over the annual cycle. They take intoaccount the optical properties of the gases and other control-ling influences, importantly temperature and its verticalstructure, spectral interference by other GHGs includingwater vapor, and interference by clouds (e.g., Ref. 25).Although based on well-understood physics, such calcula-tions are not trivial, and likewise calculation of global aver-age forcing is not trivial. Temperature exhibits seasonal andmore rapid variation, affecting greenhouse gas forcingdirectly and indirectly through radiative interactions withwater vapor and clouds. Consequently, even though theincremental amount of the gas is fairly uniformly distributedin the global atmosphere on account of the long residencetimes of these gases, forcings by incremental GHGs are, ingeneral, dependent on location, season, time of day, cloudi-ness, and the like. All these situational dependences must beaccounted for in calculating the global averages of the sev-eral forcings. Still more complicated is modeling of forcingsby tropospheric aerosols, whose properties and distributionsin the atmosphere are highly nonuniform and whose interac-tions with clouds are not yet well understood. It is estimatesof forcings obtained in this way, together with uncertaintiesinferred from inter-model spread and observational con-straints, that were examined in the 2013 IPCC Assessment4

and which are summarized in Fig. 2.

C. Adjustment times of forcing agents

A further important property of forcing agents (green-house gases, aerosols) influencing their impacts on climatechange is their residence times in the atmosphere. These resi-dence times are governed by chemical, physical, and, forCO2, biological processes that remove the substances fromthe atmosphere. Removal of greenhouse gases methane,nitrous oxide, and chlorofluorocarbons takes place largely bygas-phase chemical reactions on time scales of a decade orso (CH4) to a century or so (N2O, CFCs), Table II. Aerosolparticles are removed from the atmosphere mainly by precip-itation and, especially for larger particles (diameter � 3 lm)gravitationally and/or and inertially induced deposition tothe surface, on a much shorter time scale, about a week. Thesituation with anthropogenic CO2 is more complicated,because of the abundant natural background of CO2 withsources and sinks that greatly exceed the source strength of

650 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 650

anthropogenic CO2. It is thus necessary to distinguish themean atmospheric residence time of CO2, from what istermed the adjustment time of incremental CO2, the timeconstant that would characterize the rate of removal of incre-mental CO2 in the atmosphere relative to the preindustrialamount (e.g., Ref. 4, Glossary, p. 1457). This adjustmenttime is the quantity that characterizes the persistence of theintensified greenhouse effect of anthropogenic emissions ofCO2. The role of natural sources and sinks of CO2 is mani-fested in the annual fluctuations of atmospheric CO2 mixingratio seen, for example, in the Mauna Loa record, Fig. 1.From estimates of the rate of uptake of CO2 into the terres-trial biosphere and ocean (Ref. 5, Figure 7.3) the mean resi-dence time of atmospheric CO2 (amount in the atmospheredivided by removal rate) is estimated at about 3 years.However, the residence time reckoned in this way is inappro-priate for consideration of the adjustment time of incremen-tal CO2. From its definition, it is seen that the adjustmenttime of incremental CO2 is a hypothetical quantity, as it can-not be directly measured. However, it can be inferred bymodeling of the planetary carbon cycle26,27 or from consider-ations of the budget of excess atmospheric CO2, (Refs. 20and 21; SN3). Budget considerations indicate an adjustmenttime for CO2 at present of about 45 years, increasing some-what from 1958 to the present and thus likely to continue toincrease in the future.

16. “A three-dimensional model of atmospheric CO2 trans-port based on observed winds: 1. Analysis of observa-tional data,” C. D. Keeling et al., in Aspects of ClimateVariability in the Pacific and the Western Americas,edited by D. H. Peterson, Geophysical Monograph Vol.55 (American Geophysical Union, Washington D.C.,1989), pp. 165–236.http://onlinelibrary.wiley.com/doi/10.1029/GM055p0165/summary Essential introductionto interpretation of increases of CO2 over theAnthropocene. (I)–(A)

17. “An unexpected and persistent increase in global emis-sions of ozone-depleting CFC-11,” S. A. Montzka, G. S.Dutton, P. Yu, E. Ray, R. W. Portmann, J. S. Daniel, L.Kuijpers, B. D. Hall, D. Mondeel, C. Siso, and J. D.Nance, Nature 557, 413–417 (2018). https://www.nature.com/articles/s41586-018-0106-2. Decrease in CFCs isless than expected due to chemical removal, implyingpersistent sources. (E)

18. “New estimates of radiative forcing due to well mixedgreenhouse gases,” G. Myhre, E. J. Highwood, K. P.Shine, and F. Stordal, Geophys. Res. Lett. 25,2715–2718 (1998). Widely cited expressions for depen-dence of global mean forcings on mixing ratios. (I)

19. “Radiative forcing of carbon dioxide, methane, andnitrous oxide: A significant revision of the methane radi-ative forcing,” M. Etminan, G. Myhre, E. J. Highwood,and K. P. Shine, Geophys. Res. Lett. 43(12), 614–623(2016). Presents updated estimates of expressions forglobal mean forcings given in Ref 39.

20. “Control of fossil-fuel particulate black carbon andorganic matter, possibly the most effective method ofslowing global warming,” M. Z. Jacobson, correction toJ. Geophys. Res. 110, D14105 (2005). Empirical evalua-tion of adjustment time of excess atmospheric CO2. (E).

21. “The exponential eigenmodes of the carbon-climate sys-tem, and their implications for ratios of responses to for-cings,” M. R. Raupach, Earth Syst. Dyn. 4, 31–49

(2013). https://www.earth-syst-dynam.net/4/31/2013/esd-4-31-2013.html. Lucid exposition of budgets andadjustment times of excess atmospheric CO2 under dif-fering assumptions. (I).

22. “Radiative forcing by well-mixed greenhouse gases:Estimates from climate models in the IntergovernmentalPanel on Climate Change (IPCC) Fourth AssessmentReport (AR4),” W. D. Collins et al., Geophys. Res. 111,D14317 (2006). Detailed examination of radiation trans-fer models. (A)

23. “Cloud adjustment and its role in CO2 radiative forcingand climate sensitivity: A review,” T. Andrews, J. M.Gregory, P. M. Forster, and M. J. Webb, Surv. Geophys.33, 619–635 (2012). https://link.springer.com/article/10.1007/s10712-011-9152-0. Influential paper examin-ing short-term influences of incremental greenhousegases on atmospheric structure. (I)

24. “Adjustments in the forcing-feedback framework forunderstanding climate change,” S. C. Sherwood, S.Bony, O. Boucher, C. Bretherton, P. M. Forster, J. M.Gregory, and B. Stevens, Bull. Am. Meteorol. Soc. 96,217–228 (2015). Good exposition of forcing, adjustedforcing, sensitivity, feedback concepts. (I)

25. “The role of long-lived greenhouse gases as principalLW control knob that governs the global surface temper-ature for past and future climate change,” A. A. Lacis, J.E. Hansen, G. L. Russell, V. Oinas, and J. Jonas, TellusB 65, 19734 (2013). Accessible treatment of greenhouseeffect of long-lived gases and water vapor. (I)

26. “Carbon dioxide and climate impulse response functionsfor the computation of greenhouse gas metrics: A multi-model analysis,” F. Joos et al., Atmos. Chem. Phys. 13,2793–2825 (2013). Review of impulse-response modelsfor incremental CO2. (I)

27. “Long-term climate change commitment and reversibil-ity: An EMIC intercomparison,” K. Zickfeld et al.,J. Clim. 26, 5782–5809 (2013).

IV. CLIMATE SYSTEM RESPONSE TO FORCING;

CLIMATE SENSITIVITY

The key question facing the climate change research com-munity is how much global temperature would be expectedto increase for a given increase in CO2 or other GHGs in theatmosphere. Closely related is attribution of the observedincrease in global temperature to the increase in GHGs andwhether the magnitude of this increase is consistent withexpectation. Yet another key question is the time scale overwhich such temperature response would occur. Change inglobal mean surface temperature (GMST) is commonlytaken as the principal indicator of climate change under theassumption that changes in other components of the climatesystem would scale linearly with change in GMST. Thishypothesis is borne out for global-annual quantities, such asprecipitation amount, in studies with global climate models.Computer representations of the key processes of the climatesystem permit examination of the consequences of perturba-tions in amounts of greenhouse gases and aerosols or otherperturbations—(realistic or intentionally unrealistic) such aschanging the solar constant. Locally, and over shorter timeperiods, changes can be quite nonlinear, especially thoseassociated with the ice-liquid water transition. Here the focus

651 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 651

is restricted to GMST as an indicator of climate change on aglobal scale.

Recognition that a forcing would be expected to induce achange in global temperature leads to definition of a quantitycommonly denoted as Earth’s equilibrium climate sensitiv-ity, the amount by which GMST would change, for longtimes, in response to a sustained global mean forcing F, nor-malized to that forcing

Seq �DTs

For equivalently DTs ¼ SeqF: (1)

Equation (1) may also be viewed an explicit statement of thelinearity of response of GMST to forcing, at least to firstorder in the perturbation; see SN4. Determination of this so-called equilibrium sensitivity is a longstanding goal andchallenge to the climate research community. It is importanthere to qualify use of the word “equilibrium” in the contextof climate sensitivity. It should be emphasized Earth’s cli-mate is by no means an equilibrium system. A state ofdynamic equilibrium is characterized by detailed balance(equal and opposite fluxes) on all paths. This condition isdecidedly not met by Earth’s climate system, which is char-acterized by shortwave radiation in and longwave radiationout. Nonetheless the term “equilibrium climate sensitivity”is in widespread use; hence the subscript “eq.”

Importantly, the equilibrium sensitivity is pertinent to thechanges in surface temperature that characterize the responseof Earth’s climate system to perturbations generally, not justto the hypothetical situation of a new steady state. Let the Nbe the net energy flux into the system. For the unperturbedsystem, this net energy flux consists of two terms, theabsorbed shortwave flux Q0 and the emitted longwave fluxE0, where the subscripts 0 denote the initial, unperturbedstate. In this initial state

N0 ¼ Q0 þ E0 ¼ 0; (2)

cf. Eq. (1) of Resource Letter GECC-1. The response of theclimate system to a perturbation is illustrated schematicallyin Fig. 4 for the example of a hypothetical step-function per-turbation, the application of a constant forcing Fapp com-mencing at time t0, shown in the bottom panel. Here,“external” means that the perturbation is not a part of the cli-mate system, for example, the addition of an incrementalamount of a greenhouse gas into the atmosphere or, hypo-thetically, an increase in the solar constant. The change inradiation budget due to the forcing is to be distinguishedfrom the change in the budget that would result from climatesystem response to the perturbation. With the sign conven-tion that positive forcing F denotes an increase in the netenergy flux into the system, then initially, at time immedi-ately after t0, the rate of change of the heat content of theplanet, N, previously 0, is abruptly increased by the forcing(middle panel of Fig. 4),

Nðt0þÞ ¼ Q0 þ E0 þ Fapp ¼ Fapp: (3)

The positive net energy flux results in a gradual increase inthe heat content of the planet and in turn the surface tempera-ture, top panel. As the temperature increases, the climate sys-tem responds by increasing the emitted longwave radiation(decreasing E; recall the sign convention that E is the nega-tive of the emitted longwave flux) and/or by increasing

reflected shortwave irradiance from the planet (decreasing Q,the absorbed shortwave flux). Both responses would, overtime, decrease the net downward radiative flux at the TOA,N, offsetting the imposed forcing. This response of the cli-mate system to an imposed forcing is the reason that aftertime, the energy imbalance of the planet is much less thanthe applied forcing. A simple, but considerably more realis-tic, representation of the time response of the climate systemto perturbations is given by a two-compartment model of theclimate system.28–32 This model exhibits two time constants,a short time constant of about 8 years characteristic of theresponse of the ocean mixed layer and a long time constantof about 500 years, characteristic of the response of the deepocean.

Under the assumption (SN4) that the decrease in netdownward flux at the TOA is linear in the change in surfacetemperature, then over time subsequent to imposition of theperturbation

N t > 0ð Þ ¼ Fapp þ@Q

@Ts

����0

þ @E

@Ts

����0

!DTs tð Þ; (4)

where the subscript 0 denotes that the derivatives are evalu-ated at the initial state, consistent with treatment of effect ofthe perturbation to first order. At long time such that a newsteady state is reached, the net flux into the systemNðt!1Þ ! 0. At such time

DTs 1ð Þ ¼ �@Q

@Ts

����0

þ @E

@Ts

����0

!�1

Fapp (5)

so that the identification can be made, cf. Eq. (1)

Seq ¼ �@Q

@Ts

����0

þ @E

@Ts

����0

!�1

: (6)

Equation (6), which formally relates the equilibrium cli-mate sensitivity to the derivatives of absorbed shortwaveflux and emitted longwave flux with respect to surface

Fig. 4. Conceptual depiction as a function of time of response of climate

system to step-function forcing at time t0. Fapp is the applied forcing; DTs is

the change in surface temperature; N is the rate of change of heat content

with time or, equivalently, the net radiative flux imbalance at the top of the

atmosphere, TOA.

652 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 652

temperature, is useful in defining feedbacks in the climatesystem and as a tool for examining contributions to climatesensitivity in three-dimensional global climate models.The feedback terminology, which derives from its use inanalyzing electronic circuits, was introduced into the cli-mate change literature by Hansen et al.33 An importantinsight from Fig. 4, and the result holds generally, is thatbecause of climate system response to forcing, a forcing isnot a quantity that can be determined by measurement, forexample, by measurement of the energy imbalance at theTOA. Rather it is a quantity that can be obtained only bycalculation based on knowledge of the perturbation inatmospheric composition, taking into account the state ofthe atmosphere.

The identification in Eq. (6) together with Eq. (4) showsthat the equilibrium sensitivity is a useful measure of climatechange not just when the system has attained a new steadystate, but generally (e.g., Ref. 34). Solving Eq. (4) for DTs(t)yields

DTsðtÞ ¼ Seq FðtÞ � NðtÞ½ �; (7a)

where DTs(t) and F(t) are defined relative to an initial steadystate. Still more generally, for two states of the climate sys-tem, neither of which is at steady state

DTs ¼ SeqðDF� DNÞ; (7b)

where all D’s are calculated between the two climate states.These expressions, which rest on conservation of energy inthe climate system and which therefore might be considered“the simplest imaginable parameterization of climate systemresponse to radiative forcing,”35 are central to interpretationof climate change. In contrast to forcing, the energy imbal-ance of the climate system N is a quantity that can be mea-sured, in principle at the TOA from satellites and in practiceas the rate of change of global heat content, the great major-ity of which is manifested in change in temperature of theglobal ocean.36

An initial indication of the equilibrium sensitivity ofEarth’s climate to a radiative perturbation may be gainedfrom the model of the climate system without feedbacksgiven by Eqs. (1)–(3) in Resource Letter GECC-1, §II. In theabsence of feedbacks, that is, neither the planetary co-albedo, c, nor the effective emissivity, e, (Resource LetterGECC-1, Sec. II) is a function of surface temperature, Ts, thederivatives in Eq. (4) are

@Q

@Ts

¼ 0;@E

@Ts

¼ �4erT3s ¼ �

4

Ts

erT4s ¼ �

4

Ts

cJS

4

¼ � cJS

Ts

; (8)

from which the sensitivity of this no-feedback climate sys-tem, denoted by the subscript NF, is

SNF ¼Ts

cJS

: (9)

This no-feedback sensitivity is the equilibrium sensitivity ofa Stefan-Boltzmann–like planet that is gray in both the short-wave (co-albedo c¼ 0.71) and the longwave (emissivitye¼ 0.62) but with both grayness quantities held constant is

about 0.30 K/(W�m�2). If the assumptions of this modelwere accurate for Earth, we would have solved a big part ofthe science question having to do with climate change.Unfortunately, the problem is much more complicated thanthis because the actual response of Earth’s climate system toperturbations includes the effects of feedbacks.

Historical understanding of the magnitude of Earth’sactual “equilibrium” climate sensitivity is summarized inFig. 5, which shows best estimates of this quantity and ofassociated uncertainties going back to the original estimateby Arrhenius37 (Nobel Prize in chemistry for ionic dissocia-tion; activation energy of chemical reactions). Arrheniusobtained his estimate by extensive calculations as a functionof latitude and season, taking into account snow/ice cover,relative humidity, cloudiness, and the absorption spectra ofwater vapor and carbon dioxide. Also shown for reference isthe no-feedback sensitivity just calculated, with the datebeing that of the determination of Stefan’s constant. (Thereis no indication that Stefan calculated this sensitivity, but hedid use his law to obtain a very accurate determination of thetemperature of the Sun.39) Despite much work, especiallyover the past several decades, Earth’s equilibrium sensitivityremains quite uncertain, for a variety of reasons, as summa-rized in the 2013 IPCC Assessment (Ref. 4, Sec. TFE.6). Asthe value of this sensitivity has major implications on strate-gies to constrain global temperature rise due to increases inGHG concentrations, determination of Earth’s climate sensi-tivity is the “holy grail” of current climate change research.

Attention is called to the two ordinate scales on the graphin Fig. 5. For historical reasons that go back to Arrhenius’1896 estimate,37 the sensitivity of the climate system to aradiative perturbation is commonly expressed as long-termtemperature increase DT2� that would result from a sustaineddoubling of atmospheric CO2, denoted here and elsewhere asthe CO2 doubling temperature of the planet. This quantity isalso commonly denoted the “equilibrium climate sensitivity”of the planet, but as stressed above the system is not an equi-librium system, and, moreover, and the quantity is a temper-ature change for a specific forcing, not a sensitivity.Expressing the long-term sensitivity as DT2�, denominatedon the left-hand axis of the figure, particularizes the sensitiv-ity as the response to CO2 doubling; this particularization

Fig. 5. Historical estimates of Earth’s “equilibrium” climate sensitivity.

Sensitivities are given as long-term temperature increase DT2�, left axis, that

would result in response to sustained doubling of atmospheric CO2 and in sys-

tematic units, K/(W�m�2), right axis, with the conversion (forcing due to dou-

bled CO2) taken as 3.7 W�m�2. The value denoted Stefan represents the no-

feedback sensitivity calculated, Eq. (15), using the Stefan-Boltzmann radiation

law but was not actually calculated by Stefan. The value denoted Arrhenius is

by direct calculation (Ref. 37). Charney NRC is from an assessment by the U.

S. National Research Council (Ref. 38). The several ranges (central 66% of the

likelihood distribution) and, for 2007, point and range, at right are best esti-

mates in the several reports of the Intergovernmental Panel on Climate

Change, IPCC; also shown for 2013 is central 90% range, dashed.

653 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 653

raises questions, for example, about possible dependence onthe magnitude of CO2 mixing ratio at the initial state andabout the response to a perturbation other than that of dou-bling CO2 (other magnitudes of CO2 perturbation or otherkinds of radiative forcings). The systematic unit for sensitiv-ity, K/(W�m�2), as determined in the above calculation forthe no-feedback climate sensitivity and denominated on theright-hand axis, has not gained much traction, much as itmight seem to be preferred. The conversion between the twoquantities is

DT2� ¼ F2�Seq; (10)

where F2� denotes the forcing that would result from dou-bling atmospheric CO2, commonly taken as 3.7 W�m�2

(Table II; Refs. 18 and 19). If F2� were accurately known,then converting between doubling temperature DT2�, unit K,and sensitivity Seq, unit K/(W�m�2), would pose no problem.However, the central 90% uncertainty range in the radiativeforcing by doubled CO2, as given in the most recent (2013)IPCC Assessment, [620%] (Ref. 4, Technical SummarySec. TFE4), is substantial.

All of the estimates of Earth’s equilibrium climate sensi-tivity shown in Fig. 5 and virtually all current estimates40

exceed the no-feedback sensitivity and are therefore indica-tive of positive feedback in the climate system. The best esti-mate and range denoted “NRC” comes from an assessmentconducted under the auspices of the National ResearchCouncil of the United States (NRC)38 that was informedlargely by early calculations with GCMs, general circulationmodels of Earth’s atmosphere. Although remarkable for theirtime, the models of that time were quite primitive by thestandards of today’s climate models. The several ranges and,for the 2007 assessment, best estimate and range, of the suc-cessive Assessment Reports of the IPCC are based on con-siderations of multiple sources of information: GCMcalculations, energy balance estimates, and estimates frompaleo climates. Also shown in Fig. 5 is the estimate of thecentral 90% of the likelihood distribution for climate sensi-tivity given in the 2013 Assessment, the lower end of whichcoincides with the no-feedback sensitivity.

As a summary of the present state of understanding ofEarth’s climate sensitivity, Fig. 5 might be viewed as “good-news, bad-news.” The good news is that the best estimatesgiven in the several studies and assessments are all about thesame. The bad news is that the present estimate of this sensi-tivity is uncertain to a factor of 3, where the uncertaintyrange represents the central 66% of the likelihood distribu-tion, approximately 61r. Further bad news is that despite allthe intervening research, the estimates of the uncertaintyrange have not diminished between the 1979 NRC reportand the 2013 IPCC report. This uncertainty thus remainsquite large relative to the accuracy, say 25%, required todevelop emission control strategies to limit the futureincrease in GMST to a desired value.

An important point pertinent to quantifying past and pro-spective climate change has to do with the small magnitudesof the changes in radiative fluxes and global temperature rel-ative to the magnitudes of the initial, unperturbed quantities.The observed change in GMST of about 0.8 K (Ref. 4, Sec.2.4) represents a change of about 0.3% relative to the initial287 K. Even the 2 K increase in GMST that is widely consid-ered a threshold of dangerous anthropogenic interferencewith the climate41–44 and which is the target maximum

increase in temperature specified in the 2015 ParisAgreement of the United Nations Framework Convention onClimate Change45 represents a change in GMST of less than1%. The challenge to the climate change research commu-nity is to gain quantitative understanding of the changes inquantities influencing climate change and the expectedresponse of the system to the accuracy necessary forinformed decision making regarding prospective controls onfuture emissions of climate influencing substances. Suchquantitative understanding is essential to answering “whatif” questions regarding the consequences of future emissionsof climate influencing substances. Determining the increasein GMST that would result if the mixing ratio of CO2 wereto increase to twice its preindustrial value to some desiredlevel of accuracy, say 25%, is a daunting challenge.

28. “Atmospheric CO2 and climate: Importance of the tran-sient response,” S. H. Schneider and S. L. Thompson,J. Geophys. Res. 86, 3135–3147 (1981). Insightful paperby a pioneer in the field. (E)–(I).

29. “Vertical heat transports in the ocean and their effect ontime-dependent climate change,” J. M. Gregory, Clim.Dyn. 16, 501–515 (2000). https://link.springer.com/article/10.1007/s003820000059. Important paper examiningtime scales of climate system response to perturba-tions. (I).

30. “Implications of delayed actions in addressing carbondioxide emission reduction in the context of geo-engi-neering,” O. Boucher, J. A. Lowe, and C. D. Jones,Clim. Change 92, 261–273 (2009). https://link.springer.com/article/10.1007/s10584-008-9489-7. Exposition oftwo–time-constant model and application via convolu-tion of impulse response function with prospective forc-ings. (E).

31. “Probing the fast and slow components of global warm-ing by returning abruptly to preindustrial forcing,” I. M.Held, M. Winton, K. Takahashi, T. Delworth, F. Zeng,and G. K. Vallis, J. Clim. 23, 2418–2427 (2010). http://journals.ametsoc.org/doi/abs/10.1175/2009jcli3466.1Influential study with general circulation model andsimple model. (I)

32. “Determination of Earth’s transient and equilibrium cli-mate sensitivities from observations over the twentiethcentury: Strong dependence on assumed forcing,” S. E.Schwartz, Surv. Geophys. 33, 745–777 (2012). https://link.springer.com/article/10.1007/s10712-012-9180–4.Analysis using two-compartment climate model. (E) - (I).

33. “Climate sensitivity: Analysis of feedback mechanisms,”J. Hansen, A. Lacis, D. Rind, G. Russell, P. Stone, I.Fung, R. Ruedy, and J. Lerner, in Climate Processesand Climate Sensitivity, edited by J. E. Hansen and T.Takahashi, AGU Geophysical Monograph 29. AmericanGeophysical Union, pp 130–163 (1984). https://doi.org/10.1029/GM029p0130. Accessible; highly influentialpaper. (I)

34. “Inference of climate sensitivity from analysis of Earth’senergy budget,” P. M. Forster, Annu. Rev. Earth Planet.Sci. 44 85–106 (2016). Good recent review. (E)

35. Acknowledgment for the phrase to Bjorn Stevens, MaxPlanck Institute for Meteorology, Hamburg, Germany.

36. “An imperative to monitor Earth’s energy imbalance,” K.Von Schuckmann, M. D. Palmer, K. E. Trenberth, A.Cazenave, D. Chambers, N. Champollion, J. Hansen, S. A.Josey, N. Loeb, P. P. Mathieu, and B. Meyssignac, Nat.

654 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 654

Clim. Change 6, 138–144 (2016). https://www.nature.com/articles/nclimate2876. Important recent overview. (I)

37. “On the Influence of carbonic acid in the air upon thetemperature of the ground,” S. Arrhenius, Philos. Mag.S.5 41(251), 237–276 (1896). Historically important;defined the problem of climate system response to dou-bling of CO2. http://onlinerouting.sdsu.edu/arrhenius_paper_1896.pdf. (I)

38. “Carbon dioxide and climate: A scientific assessment,”J. G. Charney, A. Arakawa, D. J. Baker, B. Bolin, R. E.Dickinson, R. M. Goody, C. E. Leith, H. M. Stommel,and C. I. Wunsch, Natl. Acad. Sci., 22 pp. Washington,DC. (1979). http://download.nap.edu/cart/download.cgi?record_id¼12181. Influential summary of state ofknowledge at the time. (E)

39. “€Uber die Beziehung zwischen der W€armestrahlungund der Temperatur,” J. Stefan, Sitz. Math. Naturwiss.Cl. Kais. Akad. Wiss. 1, 391–428 (1879). http://www.ing-buero-ebel.de/strahlung/Original/Stefan1879.pdf. Historically important discovery of a key physicallaw. (E)

40. “Beyond equilibrium climate sensitivity,” R. Knutti, M.A. Rugenstein, and G. C. Hegerl, Nat. Geosci. 10,727–744 (2017). https://www.nature.com/articles/ngeo3017.Compilation of estimates of climate sensitivity (E).

41. “Community Strategy on Climate Change - CouncilConclusions,” Environment Council, Council of theEuropean Union, 1996. http://europa.eu/rapid/press-release_PRES-96-188_en.htm. Perhaps the earliestinternational agreement asserting “that global averagetemperatures should not exceed 2 degrees above pre-industrial level.” Among the signatories was AngelaMerkel in her capacity as Minister of the Environmentof Germany. (E).

42. “Report of the Conference of the Parties on its fifteenthsession, held in Copenhagen from 7 to 19 December2009. Addendum Part Two.” United Nations FrameworkConvention on Climate Change. (2010). http://unfccc.int/resource/docs/2009/cop15/eng/11a01.pdf. Historicallyimportant. (E).

43. “Dangerous anthropogenic interference, dangerous cli-matic change, and harmful climatic change: non-trivialdistinctions with significant policy implications,” L. D.Harvey, Clim. Change 82, 1–25 (2007). https://link.springer.com/article/10.1007/s10584-006-9183-6. Pedagogical. (E)

44. “Increasing impacts of climate change upon ecosystemswith increasing global mean temperature rise,” R.Warren, J. Price, A. Fischlin, S. de la Nava Santos, andG. Midgley, Clim. Change 106, 141–177 (2011). https://link.springer.com/article/10.1007/s10584-010-9923-5.Assessment. (E)

45. “Paris Agreement,” United Nations FrameworkConvention on Climate Change, https://unfccc.int/files/meetings/paris_nov_2015/application/pdf/paris_agreement_english.pdf. Historically important. (E)

V. INFERENCES AND IMPLICATIONS

Knowledge of the forcing over the Anthropocene togetherwith Eq. (7b) relating forcing to expected change in globalmean surface temperature (GMST) serves as the basis forcomparison of expected and observed change in GMST. Theobserved increase in GMST over the period of globally

representative instrumental measurements from the secondhalf of the 19th century to 2011 given by the 2013 IPCCAssessment (Ref. 4, Sec. 2.4.3.) was about 0.8 K (https://cru-data.uea.ac.uk/cru/data/temperature/HadCRUT4-gl.dat). Thebest estimate total forcing over the period 1850–2011 is2.2 W�m�2 (Fig. 2). The present planetary heating rateinferred from increase in the heat content of the planet overtime, N, also taken from the 2013 IPCC Assessment (Ref. 4,Box 3.1) is about 0.5 W�m�2. For the equilibrium climatesensitivity expressed as CO2 doubling temperature DT2�taken as the central value of the 2013 IPCC estimate, 3 K,and the forcing of doubled CO2 taken as 3.7 W�m�2, the cal-culated increase in GMST, 1.4 K, is substantially greater thanthe observed increase. This discrepancy is shown graphicallyin Fig. 6, which shows the increase in GMST calculated byEq. (7b) with N¼ 0.5 W�m�2 as a function of total forcingand equilibrium climate sensitivity (expressed as DT2�). Thecalculated increase in GMST, 1.4 K, falls well to the right ofthe observed value DT¼ 0.8 K denoted by the thick contourline. However, the discrepancy is readily resolved within pre-sent uncertainties in forcing and climate sensitivity as the con-tour denoting the observed increase in GMST 0.8 K can beachieved by a wide-ranging mix of forcing and climate sensi-tivity within their respective uncertainties denoted by the verti-cal and horizontal pairs of lines. Thus the observed increase inGMST can be achieved for DT2� at the low end of the rangegiven by the 2013 IPCC Assessment, 1.5 K, if the total forcingis 2.5 W�m�2 or, alternatively, for DT2� at the high end of theIPCC range, 4.5 K, if the total forcing is 1.1 W�m�2, or any-where in between. Indeed, if the forcing is at the high end ofthe IPCC range, 3.3 W�m�2, DT2� must be 1 K, a very lowsensitivity, essentially equal to the no-feedback sensitivity(Fig. 5). On the other hand, if the forcing is at the low end ofthe uncertainty range, 1.3 W�m�2, then DT2� must be 5 K,quite a high sensitivity. This analysis shows that the presentuncertainty in total forcing precludes observationally con-straining climate sensitivity, and vice versa.

Fig. 6. Dependence of expected increase in global mean surface tempera-

ture (accounting for global heating rate taken as 0.5 W�m�2; Ref. 5, Box 3.1)

on total forcing over the Anthropocene, abscissa, and equilibrium climate

sensitivity expressed as long-term temperature increase DT2� that would

result in response to sustained doubling of atmospheric CO2, ordinate.

Vertical black lines denote central 90% of the likelihood distribution of forc-

ing and horizontal lines denote central 66% of the likelihood distribution of

climate sensitivity, both as given by the 2013 IPCC Assessment (Ref. 4).

Bold contour at 0.8 K denotes approximate value of observed increase in

GMST. Points denote expected increase in GMST due to IPCC best estimate

total forcing over the Anthropocene, 2.29 W�m�2 and by long-lived green-

house gases only, 2.82 W�m �2 (Fig. 2).

655 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 655

This situation has major implications regarding develop-ing strategies to limit increase in global temperature.Importantly, reducing fossil fuel combustion to limit CO2

emissions would also reduce emissions of aerosols and pre-cursor gases. As the atmospheric residence time of theseaerosols is quite short (weeks) relative to that of CO2 andother greenhouse gases (decades to centuries), reducing fos-sil fuel emissions would thus result in a rapid decrease of thenegative aerosol forcing. In turn, total forcing and GMSTwould increase. Under this situation, the total forcing wouldincrease, in the limit, to the present forcing by long-livedGHGs, 2.8 W�m�2, denoted “GHG” in Fig. 6. For climatesensitivity at its best estimate value, the resulting tempera-ture increase relative to preindustrial would be 1.9 K. Withinpresent uncertainty in climate sensitivity, as given by the2013 IPCC Assessment,4 the temperature increase above pre-industrial to which the planet is committed on account oflong-lived GHGs now in the atmosphere would range fromthe 1 K already realized to, at the high end of the sensitivityrange, as great as 2.8 K [Ref. 46]. The latter substantiallyexceeds the widely accepted 2 K threshold for dangerousanthropogenic interference with the climate system (Sec.IV). The possibility that the negative forcing by anthropo-genic aerosols has prevented the increase in GMST thatwould otherwise have been much greater has been character-ized as a “Faustian bargain.”47 To forestall the full impact ofthe incremental GHGs, the aerosol offset or some other formof geoengineering would need to be maintained long afterthe benefit gained from combustion of fossil fuels has beenenjoyed.

With respect to societal decision making on future emis-sions of CO2 and other greenhouse gases, the consequencesof the present uncertainties are quite discomfiting. If climatesensitivity is at the low end of the uncertainty range, thenperhaps there is time for an orderly transition from relianceon fossil fuels as the primary source of the world’s energy toalternative sources of energy. If, however, climate sensitivityis at the high end of that range, then future emissions willadd to the already committed increase in GMST that isalready well beyond the 2-K threshold that is widely consid-ered to represent the onset of dangerous interference withEarth’s climate.

46. “Unrealized global temperature increase: Implications ofcurrent uncertainties,” S. E. Schwartz, J. Geophys. Res.:Atmospheres 123 (2018). https://doi.org/10.1002/2017JD028121. Systematic examination of implicationsof uncertainties in aerosol forcing, climate sensitivity,and adjustment time of excess CO2. (E)–(I).

47. “Sun and dust versus greenhouse gases: An assessmentof their relative roles in global climate change,” J. E.Hansen and A. A. Lacis, Nature 346, 713–719 (1990).

https://www.nature.com/articles/346713a0. Importantassessment of the consequences of aerosol forcing andthe greatly differing atmospheric residence times ofaerosols and GHGs; (E)

VI. SUMMARY

This Resource Letter and the companion Resource Letter(GECC-1) have reviewed the so-called greenhouse effect oftrace species in the atmosphere, water vapor, CO2 and otherpolyatomic molecules that absorb and radiate in the thermalinfrared, and clouds, calling attention to pertinent primaryand secondary literature. The greenhouse effect is a key ele-ment of Earth’s climate system responsible for a globalmean surface temperature, average roughly 287 K, muchgreater than the radiative temperature at the top of the atmo-sphere, 255 K. Because of human activities, largely fossilfuel combustion, but including also deforestation, agricul-tural activities, and other industrial activities, amounts oflong-lived greenhouse gases in the atmosphere, haveincreased substantially relative to their preindustrial values.The radiative forcing of the incremental greenhouse gasesover the Anthropocene to date, calculation of which restsrather confidently on well-understood molecular propertiesand well characterized properties of the climate system, is oforder 1% relative to top-of-atmosphere fluxes and is smallalso relative to spatial and temporal variability of thesefluxes on a variety of scales. However even a 1% increase inglobal mean surface temperature, �3 K, would be of greatconsequence to human society and natural ecosystems.Present uncertainties in forcings and Earth’s climate sensiti-vity have enormous implications regarding interpretation ofincrease in GMST to date, committed future increases inGMST, and policy regarding future emissions of long-livedgreenhouse gases.

The physics of the greenhouse effect and more broadly ofEarth’s climate and climate change is a research field thathas attracted many good physicists and continues to presentimportant first-order challenges and research opportunities.

ACKNOWLEDGMENTS

The author is indebted to numerous friends and colleaguesfor reading and commenting on earlier drafts of this ResourceLetter. The author thanks several referees for valuablesuggestions and Mario Belloni for helpful comments andcareful editing. This study was supported by the U.S.Department of Energy’s Atmospheric System ResearchProgram (Office of Science, OBER) under Contract No. DE-SC0012704.

656 Am. J. Phys., Vol. 86, No. 9, September 2018 Stephen E. Schwartz 656