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500
100
80
450
60
400
40
E = 10,200 Tg-C/yr
E = 6281.4 Tg-C/yr
E = 0 Tg-C/yr
350
20
Low emission rate
High emission rate
300
0
1960
1980
2000
Year
250
2000
2050
2100
2150
2200
2250
2300
Year
Figure 3.12. Data-constrained e -folding lifetime of
carbon dioxide from Equation 3.21. The mean lifetime
is about
Figure 3.13. Time-dependent change in CO 2 (g)
mixing ratio from Equation 3.22 under three different
emission scenarios: (a) with 2010 emissions from fossil
fuels plus permanent deforestation of 10,200 Tg-C
yr −1 ,(b)withanemission rate derived from
E
40 years, with a rough range of 30-50
years. The dashed lines are linear fits to the data.
The two curves correspond to low and high carbon
emission rates from permanent deforestation added
to constant levels of fossil fuel emissions in each case.
Updated from Jacobson (2005c).
=
6,281.4 Tg-C yr −1 that gives a constant
mixing ratio of CO 2 (g), and (c) with zero emissions.
Other conditions are
= a (0)
/ =
b =
275 ppmv,
a (0)
=
115
ppmv, and
=
40 years.
decreasing. In addition, continued permanent deforesta-
tion since 1960 has reduced the vegetation mass avail-
able for CO 2 (g) uptake by photosynthesis. Both factors
have contributed to a greater residence time of CO 2 (g)
in the air since 1960.
The exact solution to the change in mixing ratio over
time from Equation 3.20 is
scenarios: (a) one with 2010 emissions; (b) one with
emissions reduced to E
,which forces the
anthropogenic mixing ratio of CO 2 (g) over time to
equal a constant initial anthropogenic mixing ratio; and
(c) one with zero emissions. In all cases, the lifetime
of CO 2 (g) is held to 40 years. In cases (a) and (c),
the atmospheric mixing ratio is not initially in equi-
librium with its emissions, so the atmospheric mixing
ratio increases and decreases, respectively, over time to
reach equilibrium.
Case (a) indicates that, even if E were held constant
at the 2010 level, the CO 2 (g) atmospheric mixing ratio
would increase for many years. Today, the CO 2 (g) emis-
sion rate is increasing, not staying constant. As such,
the CO 2 (g) mixing ratio will rise more in reality than
it will with the highest emission rate scenario in Figure
3.13. Cases (b) and (c) suggest that the only way to
reduce the ambient mixing ratio of CO 2 (g) is to reduce
emissions from 10,200 Tg-C yr 1
=
(0)
/
= a (0) e t / +
e t / )
a ( t )
E (1
(3.22)
where
b is the initial anthropogenic
mixing ratio (ppmv) at time t
a (0)
=
(0)
0, which corresponds to
a base year of interest. For example, if 2010 is selected
as the base year ( t
=
=
0), and if the total mixing ratio
of CO 2 (g) in 2010 is
(0)
=
390 ppmv, then
a (0)
=
390
115 ppmv. Because emission rates are
generally in units of Tg-C/yr, whereas Equation 3.22
requires emission units of ppmv-CO 2 (g)/yr, the constant
2,184.82 Tg-C/ppmv-CO 2 (g) is used to convert units
for the equation.
The solution in Equation 3.22 suggests that, in the
absence of emissions, the mixing ratio of CO 2 (g) will
decay exponentially to 1/ e its initial value when t
275
=
to less than 6,281
Tg-C yr 1 .
As illustrated in Example 3.8, it will be necessary to
reduce 2030 emissions by about 64 percent to stabilize
CO 2 (g) to 360 ppmv after 2030. The emission reduction
in 2030 represents an 80 percent reduction of 2010
anthropogenic emissions. As such, any effort to stabilize
CO 2 (g) at 360 ppmv requires at least an 80 percent
=
.
The equation also suggests that, in steady state ( t
→∞
),
the mixing ratio approaches
E .Once Equa-
tion 3.22 is solved, the new total mixing ratio of CO 2 (g)
is
a (
)
=
= b + a ( t ).
Figure 3.13 illustrates how CO 2 (g) changes accord-
ing to Equation 3.22 under three constant emission
( t )
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