According to the simplest model of the earth’s radiative balance, global warming will occur with certainty as humankind increases its production and consumption of nonsolar energy. This prediction is revisited, using a broader model that allows the greenhouse effect to be considered. The new model predicts a global warming of ΔTE=(114 K)ε, where ε is the rate of surface energy release in units of the average incident solar radiation, 342 W m−2, and ΔTE is the average temperature rise at the earth’s surface. Present values of these quantities, excluding geothermal sources, are ε=0.69×10−4 and ΔTE=7.9 mK. The model assigns a small number of optical parameters to the atmosphere and surface and qualifies the simple warming prediction: It is rigorous only if parameters other than ε are unchanged. The model is not complex and should serve as an aid to an elementary understanding of global warming.

1.
See, for example, C. Kittel and H. Kroemer, Thermal Physics (Freeman, San Francisco, 1980), 2nd ed., Exercise 4.5, p. 111.
2.
D. L. Hartmann, Global Physical Climatology (Academic, San Diego, 1994), p. 2.
3.
We adopt as our observed “surface temperature” the time- and geographically averaged lower-atmosphere value measured by satellite. This temperature is consistent with the use of elementary models that deal with a single global surface temperature, interpreted as a similarly averaged value.
4.
R. W.
Spencer
and
J. R.
Christy
, “
Precise monitoring of global temperature trends from satellites
,”
Science
247
,
1558
1562
(
1990
);
J. R.
Christy
,
R. W.
Spencer
, and
W. D.
Braswell
, “
How accurate are satellite thermometers?
Nature (London)
389
,
342
(
1997
).
5.
P. D.
Jones
and
K. R.
Briffa
, “
Global surface air temperature variations during the 20th century: I. Spatial, temporal, and seasonal details
,”
The Holocene
2
,
165
179
(
1992
).
6.
A.
Rose
, “
A global view of solar energy in rational units
,”
Phys. Status Solidi A
56
,
11
26
(
1979
);
A.
Rose
, “
Solar energy: A global view
,”
CHEMTECH
11
,
566
571
(
1981
);
A.
Rose
,
CHEMTECH
11
,
694
697
(
1981
). These papers contain almost identical material.
7.
A.
Rose
, “
World energy supply and greenhouse effect
,”
Mol. Cryst. Liq. Cryst.
175
,
159
168
(
1989
).
8.
J. B. J.
Fourier
,
Mém. de l’Ac. R. d. Sci. de l’Inst. de France, t.
vii
, (
1827
).
9.
S.
Arrhenius
, “
On the influence of carbonic acid in the air upon the temperature of the ground
,”
Philos. Mag.
41
,
237
276
(
1896
).
10.
J. T.
Kiehl
and
K. E.
Trenberth
, “
Earth’s annual global mean energy budget
,”
Bull. Am. Meteorol. Soc.
78
,
197
208
(
1997
).
11.
The greenhouse effect was apparently first recognized by Fourier (Ref. 8), whose terminology has persisted despite the fact that the atmosphere does not behave in the manner of hothouse glass.
The role of “greenhouse gases” were first treated in detail by Arrhenius (Ref. 9) who summarizes the relevant 19th-century literature.
The greenhouse effect is an implicit factor in every modern model involving radiative and other means of energy transport in the atmosphere. Kiehl and Trenberth (Ref. 10) provide a brief, readable, yet comprehensive review.
12.
H.
van Loon
and
K.
Labitzke
, “
The 10–12-year atmospheric oscillation
,”
Meteorol. Z.
3
,
259
266
(
1994
);
J.
Lean
and
D.
Rind
, “
Climate forcing by changing solar radiation
,”
J. Climate
11
,
3069
3094
(
1998
).
13.
B. A.
Tinsley
and
K. A.
Beard
, “
Links between variations in solar activity, atmospheric conductivity, and clouds: An informal workshop
,”
Bull. Am. Meteorol. Soc.
78
,
685
687
(
1997
).
14.
W. D. Sellers, Physical Climatology (University of Chicago Press, Chicago, 1965);
M. I. Budyko, The Earth’s Climate: Past and Future (Academic, New York, 1982).
15.
A. Henderson-Sellers and P. J. Robinson, Contemporary Climatology (Addison–Wesley–Longman, Edinburgh Gate, UK, 1986).
16.
R. E. Dickinson, “Impact of human activities on climate–a framework,” in Sustainable Development of the Biosphere, edited by W. C. Clark and R. E. Munn (University Press, Cambridge, 1986), Chap. 9, pp. 252–289.
17.
S. H. Schneider, “Introduction to climate modeling,” in Climate System Modeling, edited by K. E. Trenberth (Cambridge U.P., Cambridge, UK, 1992), pp. 3–26.
18.
J. W.
Firor
, “
Resource letter: GW-1: Global warming
,”
Am. J. Phys.
62
,
490
495
(
1994
).
19.
S. R.
Weart
, “
The discovery of the risk of global warming
,”
Phys. Today
50
(
1
),
34
40
(
1997
).
20.
R. C.
Willson
, “
Total solar irradiance trend during solar cycles 21 and 22
,”
Science
277
,
1963
1965
(
1997
).
21.
Texts such as Refs. 2, 14, and 15 may be consulted for the details of the atmospheric absorption spectra. In our model, effective overall absorptivities are assigned to the atmosphere as single parameters.
22.
The simplest two-layer model is discussed clearly by C. Kittel and H. Kroemer, Ref. 1, 4th printing, pp. 115–116.
23.
R. E. Dickinson, “Modeling climate changes due to carbon dioxide increases,” in Carbon Dioxide Review 1982, edited by W. C. Clark (Clarendon, Oxford, and Oxford U. P., New York, 1982), pp. 103–142.
24.
Because the temperature and density of the upper atmosphere vary greatly with altitude, an effective radiation temperature or “skin” temperature, based on the overall IR emission, is usually assigned. The value 255 K is typically quoted (Refs. 17 and 23). In our elementary model, IR originates both at the surface and in the atmosphere, and the latter therefore has a somewhat lower temperature (246 K). The number to be compared with the typical 255 K is the temperature of an ideal body emitting the total IR flux, in our standard model 228 W m−2, and therefore our TAskin=251 K.
25.
C. M. R. Fowler, The Solid Earth: An Introduction to Global Geophysics (Cambridge U.P., Cambridge, UK, 1990), p. 234. Average geothermal flux densities are continents and shelves, 57 mW m−2; oceans and basins, 99 mW m−2; surface average, 82 mW m−2.
26.
W.
Häfele
, “
A systems approach to energy
,”
Am. Sci.
62
,
438
447
(
1974
).
27.
U.S. Bureau of the Census, Statistical Abstract of the United States 1998 (USGPO, Washington, DC, 1998), 118th ed. World primary energy production appears in Table 962, p. 595.
28.
Data for 1970 are extrapolated from the more detailed 1980–1996 data in order to effectively incorporate recent data revisions and methods of defining energy categories. Our total 1970 production agrees with Häfele’s (Ref. 26) for 1970.
29.
Rose adopted the “sun” as his unit of flux density, informally choosing one sun to be 200 W m−2, the approximate flux density of incoming UV at the surface. Because this quantity varies with model parameters, we prefer to retain our own sun units with εC=ΔSC/S0, slightly smaller than two-thirds of Rose’s parameter, which he did not name.
30.
J. T. Houghton, L. G. Meira Filhon, J. Bruce, H. Lee, B. A. Callander, E. Haites, N. Harris, and K. Maskell, editors, Climate Change 1994. Radiative Forcing of Climate Change and an Evaluation of the IPCC IS92 Emission Scenarios (Cambridge U.P., New York, 1995).
31.
A. H.
Gordon
, “
Weekdays warmer than weekends?
Nature (London)
367
,
325
326
(
1994
).
32.
D. E.
Lenschow
, “
Next step for weekday warming
,”
Nature (London)
369
,
193
(
1994
).
33.
D. H.
Douglass
,
Bull. Am. Phys. Soc.
44
,
1422
(
1999
).
34.
R. C.
Balling
,Jr.
and
R. S.
Cerveny
, “
Influence of lunar phase on daily global temperatures
,”
Science
267
,
1481
1483
(
1995
);
R. C.
Balling
, Jr.
and
R. S.
Cerveny
,
Science
269
,
1285
(
1995
).
35.
J. C.
Dyre
, “
Lunar phase influence on global temperatures
,”
Science
269
,
1284
-
1285
(
1995
);
C. V.
Voorhies
, same title,
Science
269
,
1285
(
1995
).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.