Graphical Means to Understanding the Global Atmospheric Budget Robin Hood Diagrams

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Here I provide a step-by step description of each vector in a Robin Hood diagram of the global atmospheric carbon budget (Fig. 13.2). The budget and figure is constructed arbitrarily for the year 1990. This vector approach is referred to as a Robin Hood diagram because of the abundance of arrows (Inez Fung, personal communication, 2000); and the overall process of solving for land and ocean carbon sinks using 13C02 and 12C02 is known as a 'double deconvolution' (Heimann and Keeling, 1989). In the conventional representation of the global budget, observational changes (labeled with a 7 in Fig. 13.2) represent some combination of fossil fuel emissions (6), land disequilibria forcing (5), a terrestrial carbon sink (4), ocean disequilibria forcing (3), and an ocean carbon sink (2). In a more general representation of the budget (Fig. 13.2B), both gross primary production ( 7) and a return flux from the biosphere to the atmosphere (8) are separately considered (Randerson et al, 2002). In this form, it can be seen that anomalies in discrimination (Aa/,) affect GPP, and ensuing disequilibria (defined below) have the potential to affect interannual variability in

220 13. Terrestrial Ecosystems and Intemnnual Variability ofCO2 and^^CO^

Robin hood diagrams Corresponding algebra

220 13. Terrestrial Ecosystems and Intemnnual Variability ofCO2 and^^CO^

Robin hood diagrams Corresponding algebra

Figure 13.2 A Robin Hood diagram is comprised of multiple vectors, each representing a different carbon cycle process. These vectors are linked to the equations describing the atmospheric budget using numbers. Observations (labeled with a 1) are given by the left hand side of the equations. A. The conventional double deconvolution representation of the terrestrial budget splits the terrestrial biosphere into a net sink (4) and an isotopic disequilibrium term (5). B. A more general representation of terrestrial carbon fluxes considers GPP ( 7) and the return flux (respiration, fires, etc., labeled with an 8). In this representation, it can be seen that small anomalies in discrimination can affect the interpretation of the atmospheric budget because they operate on a very large vector (GPP) and are not associated with a carbon source or sink. Note the change in scale. The isotopic disequilibrium forcing of the biosphere is now explicitly included as a part of the return flux vector (R¿,). In this figure, only the fraction of GPP that remains in the biosphere for year or longer is shown (-30 PgCyr"1). Adapted from Randerson el al. (2002). Modified by permission of the American Geophysical Union.

Figure 13.2 A Robin Hood diagram is comprised of multiple vectors, each representing a different carbon cycle process. These vectors are linked to the equations describing the atmospheric budget using numbers. Observations (labeled with a 1) are given by the left hand side of the equations. A. The conventional double deconvolution representation of the terrestrial budget splits the terrestrial biosphere into a net sink (4) and an isotopic disequilibrium term (5). B. A more general representation of terrestrial carbon fluxes considers GPP ( 7) and the return flux (respiration, fires, etc., labeled with an 8). In this representation, it can be seen that small anomalies in discrimination can affect the interpretation of the atmospheric budget because they operate on a very large vector (GPP) and are not associated with a carbon source or sink. Note the change in scale. The isotopic disequilibrium forcing of the biosphere is now explicitly included as a part of the return flux vector (R¿,). In this figure, only the fraction of GPP that remains in the biosphere for year or longer is shown (-30 PgCyr"1). Adapted from Randerson el al. (2002). Modified by permission of the American Geophysical Union.

atmospheric 513C. The x-axis represents the carbon flux from each process or the observed atmospheric change (with units of PgCyr-1). The y-axis represents the 13C mass flux, expressed in terms of an isoflux (with units of PgC%oyr_1).

Considering only the x-axis, it can be seen in Fig. 13.2A that the observed atmospheric growth rate of carbon (3 PgCyr-1) is only ~l/2 of fossil fuel emissions (6PgCyr_1). Land and ocean sinks must account for this difference, yet partitioning is impossible without additional information. There is one known quantity (the difference between fossil fuel emissions and the atmospheric growth rate) and two unknowns (land and ocean sinks).

Information from the atmospheric rate of change of <51SC and differences in discrimination against i3C associated with land and ocean sinks allows for a land/ocean sink partitioning. A carbon sink associated with C3 terrestrial photosynthesis and ecosystem respiration strongly discriminates against 13C (Aat, is equal to ~19%o), acts to enrich atmospheric ¿13C, and is represented in a Robin Hood diagram by a vector with a relatively steep slope (but initially with an unknown length). In contrast, a carbon sink associated with air-sea gas exchange (Aoa is equal to ~2%o) discriminates weakly against 13C, has a relatively small effect on atmospheric ¿13C, and is represented in a Robin Hood diagram by a vector with a relatively shallow slope (and again, initially with an unknown length). Only one combination of land and ocean vectors, with their slopes determined by their respective levels of discrimination, can close the atmospheric budget for both total carbon and 13C, given known values for fossil fuel emissions and isotopic disequilibria forcing terms described below.

With the added dimension that 13C provides, there is a cost. Land and ocean net carbon sinks are not the only processes affecting the rate of change of atmospheric S13C; 13C fluxes associated with the one-way ecosystem respiration and gross primary production fluxes on land and the gross one-way sea-to-air and air-to-sea fluxes in the ocean also regulate atmospheric ¿13C and thus must be estimated prior to the double deconvolution. These fluxes, known as isotopic disequilibria forcing terms, are not well characterized and contribute to substantial uncertainties in land/ocean partitioning (Fung et al., 1997). Isotopic disequilibria arise from the time delay between entry (loss from the atmosphere) and exit (return to the atmosphere) for carbon in terrestrial ecosystems and ocean reservoirs exchanging with the atmosphere, and the relatively sudden negative perturbation in the atmospheric isotopic composition caused by fossil fuels (Quay et al, 1992; Tans et al, 1993; Fung et al., 1997; Wittenberg and Esser, 1997; Gruber et al., 1999). Other processes besides fossil fuel emissions may induce disequilibria, most notably shifts in photosynthetic discrimination over a period of decades caused by a shift in C4 vegetation (Ciais et al, 1999; Townsend et al., 2002) or changes in discrimination caused by interannual variability in climate (Randerson et al., 2002).

Background

A primary simplification that is widely employed in ecosystem, regional, and global scale budget calculations is that the product of the total carbon flux and its isotopic composition, in 8 notation is a mostly conservative tracer of the 13C flux (and is known as the isoflux). For example, with the Keeling plot approach it is possible to derive the form of the linear regression relating air mass changes to the isotopic source composition of the flux using this notation or more formally by separately considering the 12C and 13C fluxes (Pataki et al, 2003). Biosphere-atmosphere trace gas exchange is quantitatively described by the following two equations:

dC2Q) 12„ i2, dt and d{UCbRh) = aabRauFab + abaRbl2Fba (13.2)

dt where l2Fab and 1 ~Fba at the global scale are the one-way fluxes of gross primary production and a return flux that is a combination of ecosystem respiration, fires, volatile organic carbon emissions, and other losses. Here a flux out of the surface is defined with a positive sign (12.F/,a) and a flux into the surface is defined with a negative sign (i2Fab). Ra and R^ represent the isotope ratios (13C/12C) of the atmosphere and biosphere, respectively, and aba and aab are the fractionation factors associated with the one-way fluxes (Criss, 1999).

Together, Eqs 13.1 and 13.2 can be approximated by the combination of the following two equations (Tans et al., 1993; Fung et al, 1997):

dt and d(SbCb)

dt where in Eq. 13.3, the mass of 12C (from Eq. 13.1) is now approximated by that for total carbon. Equation 13.4, which represents the 13C mass flux anomaly, requires the additional approximation that the isotopic composition of the photosynthesis flux in S notation is equal to the difference between the background atmosphere (Sa) and discrimination (Aab), where Aab is defined according to Farquhar et al. (1989). Aab is a positive number, typically between 16 and 21 %o for C3 plants and between 4 and 5%o for C4 plants. For many regional and global scale analyses, it is also assumed that discrimination against 13C associated with respiration and the return flux is minimal, and so the isotopic composition of the biosphere to atmosphere flux is equal to the isotopic composition of the biosphere (Sb). This does not mean, however, that Sb is necessarily equal to Sa — Aab, because carbon released today as respiration was fixed at an earlier time with a different Sa (and also with a potentially different Aab).

Observations

The observed atmospheric growth rate of carbon was approximately 3PgCyr~ in 1990. Change in the 13C mass of the atmosphere over time is represented by the time derivative of the product of the total mass and its ¿'I3C. Using the Product Rule, this derivative can be split into two components:

dt dt dt where the first term on the right hand side represents the change of 13C mass caused by the change in the total carbon inventory of the atmosphere, and the second term represents the change of 13C mass caused by the change in atmospheric <513C. In 1990, both terms contributed substantially to the 13C mass anomaly (3PgCyr_1 x -7.8%o = -23.4PgC%0yr^1 and 760 PgCyr-1 x -O^fooyr^1 = -lS.OPgCrooyr"1).

Fossil Fuels

Fossil fuels represent the single largest perturbation to the atmospheric budget on an annual basis, for both total carbon and its isotopic composition (¿13C). Without CO2 exchange with ocean or land carbon reservoirs, the observed decrease in atmospheric <513C would have been —0.16%o yr-1 in 1990, about 8 times greater than that observed (—0.02%oyr_1). Cumulatively, if all of the ~220Pg C of fossil carbon emitted prior to 1990 had remained in the atmosphere (again without ocean or land exchange) the atmosphere in 1990 would have had a 513C of -12.5%o, about 4 times greater than the observed perturbation (—7.8%o) as compared with the preindustrial atmosphere (—6.3%o).

In Fig. 13.2, I assumed that fossil fuel emissions were ôPgCyr"1 in 1990, had a global S13C signature of —28%o, and thus an isoflux of —168 Pg C%o yr-1 (Andres et al., 2000). A quantitative analysis of the uncertainties in the S13C content of fossil fuels remains to be done, and it is unlikely that this term is known to within ±l%o. At higher resolution temporal and spatial scales, even greater uncertainty exists, primarily from variability in the use of natural gas products that have a S13C of —40%o to -60%0 (Andres et al, 2000).

Terrestrial Isotopic Disequilibria

Even without a net carbon sink, gross exchange with the terrestrial biosphere enriches the contemporary atmosphere in ¿13C. Before the industrial revolution, Sa was relatively constant for many hundreds of years (Francey et al, 1999). During this time, 13C fluxes into and out of the terrestrial biosphere were roughly at steady state: enrichment of the atmosphere by photosynthesis was balanced by depletion of the atmosphere by a return flux on annual and decadal timescales (Fig. 13.3). Since the eighteenth century, however, the <513C of flux entering the biosphere has steadily decreased

224 13. Terrestrial Ecosystems and Interannual Variability of CO2 and^^COy

Calculating the terrestrial isotopic disequilibrium induced by fossil fuels

Total carbon: NPP

13C anomaly (isoflux): NPP(<Sa-Aab)

Total carbon: NPP

13C anomaly (isoflux): NPP(<Sa-Aab)

R-^CbAVt

Cb-<5b(t) = Cb-<5b(t-1 ) + NPP(<Sa-Aab) -Cb-Sb(t-t)h

R-^CbAVt

Cb-<5b(t) = Cb-<5b(t-1 ) + NPP(<Sa-Aab) -Cb-Sb(t-t)h

Figure 13,3 Steady-state fluxes for the preindustrial biosphere for both total carbon and the 13C mass flux anomaly (isoflux). Preindustrial Sa was assumed to be — 6.3%o and global A was assumed to be —16%o, reflecting a combination of C3 and C4 ecosystems. Equations are also provided for describing the time rate of change of total carbon and the mass anomaly under a changing Sa.

because of the fossil fuel perturbation to the atmosphere (Keeling et at, 1979; Francey et al., 1999). The return flux is also decreasing, but with an additional delay that reflects the residence time of carbon in plant, litter, and soil pools. The difference between the <513C of the return flux (that was fixed by photosynthesis in previous years) and a hypothetical flux that would be in equilibrium with the contemporary atmospheric state (i.e., that had a value of Sa — Aaj), is known as the isotopic disequilibrium.

In general the longer the carbon residence time, the larger the isotopic disequilibrium (Randerson et at, 1999). The maximum possible value of the disequilibrium is the difference between the preindustrial atmosphere and the contemporary atmosphere. In 1990 this maximum was ~1.6%o. For this maximum to occur, all of the carbon in the return flux would have to have an age greater than 2-3 centuries. Given the relatively large allocation of net primary production (NPP) to fine roots and leaves, most estimates of the terrestrial disequilibrium are considerably smaller, in the order of 0.2 to 0.5%o when defined with respect to NPP (Thompson and Randerson, 1999).

From the perspective of modeling or measuring isotopic disequilibria caused by a changing Sa, it is worth noting several points. First, most of the carbon that enters an ecosystem via photosynthesis leaves quickly via autotrophic respiration and leaf and fine root turnover (Trumbore, 2000). Thus the magnitude of the isotopic disequilibrium for a given year is highly sensitive to changes in Sa in the 10 or so preceding years. Second, an isotopic disequilibrium induced by a changing 8a does not depend on Aab, but only on the residence time of carbon in the surface reservoir (with the residence time defined with respect to the atmosphere). Finally, any estimate of the isotopic disequilibrium must be made in reference to a flux with a defined magnitude. For example, the isotopic disequilibrium in an ecosystem associated with NPP would be considerably greater than one defined for the same ecosystem relative to GPP. The reason for this is that while GPP is a much greater flux, much of GPP rapidly returns to the atmosphere via plant respiration and thus has a very short residence time.

The product of the isotopic disequilibrium and the gross flux is known as an 'isotopic disequilibrium forcing.' Estimates of the global terrestrial disequilibrium forcing require spatially distributed estimates of both GPP and the age distribution of carbon in ecosystem respiration, volatile organic carbon emissions, and fires. In turn, the age distribution of carbon requires knowledge of the distribution of plant functional types over the land surface, allocation, lifetimes of different plant tissues, decomposition rates, and the disturbance regime. In Fig. 13.2,1 assumed that the isotopic disequilibrium forcing from the terrestrial biosphere was 20PgC%oyr-1, following from Randerson et al (2002).

Using the single reservoir model of the terrestrial biosphere described in Fig. 13.3, in Table 13.1 I show the components of the budget required to calculate the isotopic disequilibrium and isotopic disequilibrium forcing arising from the fossil fuel perturbation to 8a. These simplified estimates are derived for a terrestrial biosphere with an NPP of 60 PgCyr^1 and a smsile carbon pool with a turnover time of 20 years. This calculation is easy to reproduce in a spreadsheet program, once a continuous time history of Sa has been constructed from observations, such as those of Francey et al. (1999).

A second class of disequilibria arises from changes in A,,/, through time (and not changes in 8a). On weekly to interannual timescales, changes in drought stress can drive trends in A ab, while over a period of decades to centuries long-term shifts in vegetation or climate are likely to be the most important drivers.

Terrestrial Carbon Sink

In the double deconvolution inversion, the magnitude of the terrestrial carbon sink is one of two unknowns. The slope of the vector is set by A ab (Fig. 13.2). Only one combination of lengths of the ocean and land sink vectors can match the observed vector, once fossil fuel and isotopic disequilibrium vectors are known.

The slope of the terrestrial carbon sink vector is proportional to Aaf,. If the terrestrial carbon sink is distributed in proportion to productivity in C3 and c4 ecosystems, then a GPP-weighted discrimination (one that combined c3 and c4 discrimination values) would be appropriate for use in a global analysis (Still et al., 2003a). If, however, most of the terrestrial carbon sink is in forests recovering from natural or human disturbance,

Table 13.1 Changes in the Terrestrial Isotopic Disequilibrium Forcing Caused by Fossil Fuel Emissions

Isotopic

Total carbon mass 13C mass Isotopic disequilibrium and fluxes1 anomaly (isoflux)2 disequilibrium forcing

Isotopic

Total carbon mass 13C mass Isotopic disequilibrium and fluxes1 anomaly (isoflux)2 disequilibrium forcing

Year

(%o)

(PgCyr^1)

(PgC)

Rh = Cb/r (PgCyr"1)

NPP • (Sa ~ Aab) (PgC %o yr—' )

C-b ' Sb (PgC%0)

(Cb-Sb)/r (PgC %o yr-1 )

h = (Rh ■ Sb)/Rh (%o)

(%o)

NPP • (Sf - Sa) (PgCXoyr"1)

1765

-6.31

60

1200

60

-1339

-26 771

-1339

-22.3

0.00

0.0

1865

-6.57

60

1200

60

-1354

-27022

-1351

-22.5

0.05

3.3

1965

-7.17

60

1200

60

-1390

-27 490

-1374

-22.9

0.28

16.8

1990

-7.79

60

1200

60

-1427

-28 065

-1402

-23.4

0.42

25.3

'The turnover time of carbon in terrestrial biosphere reservoir was assumed to be 20 years (r = 20years). Total NPP was assumed to remain at steady state throughout the model run.

2A0j was assumed to be 16%o, representing a combination of C3 and C4 vegetation. This value does not affect the magnitude of the isotopic disequilibrium or the isotopic disequilibrium forcing.

'The turnover time of carbon in terrestrial biosphere reservoir was assumed to be 20 years (r = 20years). Total NPP was assumed to remain at steady state throughout the model run.

2A0j was assumed to be 16%o, representing a combination of C3 and C4 vegetation. This value does not affect the magnitude of the isotopic disequilibrium or the isotopic disequilibrium forcing.

then this vector should have a slope reflecting primarily C3 vegetation (Still et al, 2003a). At the ecosystem scale, estimates of A„/, derived from Keeling plot observations of nighttime respiration may also be offset from the A „/, associated with a net annual carbon sink because long-term accumulation of carbon within an ecosystem occurs primarily within wood, coarse woody debris, and slowly turning-over soil organic matter pools, whereas nighttime respiration consists largely of recently fixed sugars and carbohydrates that have not undergone the additional biochemical synthesis steps required for the formation of cellulose or lignin.

Ocean Disequilibrium Forcing

This term is analogous to the terrestrial biosphere disequilibrium forcing term, with a magnitude approximately 2-3 times larger than that for the terrestrial biosphere. The gross flux from the atmosphere to the ocean is ~80PgCyr_1, based on radiocarbon and chlorofluorocarbon tracer constraints (Wanninkhof, 1992). Much of the carbon in the return flux, from the ocean to the atmosphere, has a residence time within the oceans of years to decades, and thus has an isotopic composition that is more enriched than that expected for a flux in isotopic equilibrium with the contemporary atmosphere (Gruber and Keeling, 2000). Estimates of the ocean isotopic disequilibrium forcing for the 1980s and early 1990s range between 37 Pg C%o yr-1 and 77 Pg C%o yr-1 (Tans et al, 1993; Francey et al, 1995; Gruber and Keeling, 2000) and critically depend on the relationship between wind speed and rates of air-sea gas exchange rate. In Fig. 13.2, I assumed an ocean disequilibrium forcing of 54 Pg C%o yr-1 (Gruber and Keeling, 2000).

Ocean Carbon Sink

As with the terrestrial carbon sink, the length of ocean carbon sink vector is one of two unknowns that must account for the difference between the observed atmosphere, and fossil fuel and isotopic disequilibria terms. The slope of the ocean vector is fixed from measurements of discrimination associated with the one-way ocean-to-atmosphere, and atmosphere-to-ocean fluxes (Zhang et al, 1995).

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