Tellus (1993), 45H, 409-425
Printed in Belgium - all rights reserved

Copyright (c) Munksgaard, 1993
TELLUS
ISSN 0280-6509

Balancing the carbon budget. Implications for projections of future carbon dioxide concentration changes

Climatic Research Unit
University of East Anglia
Norwich NR4 7TJ, UK

(Manuscript received 18 September 1991; in final form 12 August 1993)

ABSTRACT

1. Introduction

If one considers only the main sources (industrial emissions and fluxes related to land-use changes) and assumes that the ocean is the only sink, then the budget expressed by eq. (1) appears not to balance. This problem is exemplified using 1980s-mean data from the Intergovernmental Panel on Climate Change (IPCC; Watson et al., 1990): 5.4 +/- 0.5 GtC/yr for industrial emissions (IPCC gave the 1987 value of 5.7 GtC/yr) and 1.6 +/- 1.0 GtC/yr for emissions due to land-use changes. Since the current rate of atmospheric concentration build-up has been equivalent to around 3.4 GtC/yr over the 1980s (uncertainty +/- 0.2 GtC/yr),the residual sink averaged over this decade must be 3.7 +/- 1.7 GtC/yr.

The central value of this range is much larger than the IPCC estimate of the mean ocean flux over the 1980s, 2.0 +/- 1.0 GtC/yr (Watson et al., 1990, p. 17), implying the existence of an additional (i.e., "missing") sink. If the ocean flux were as low as 1 GtC/yr, as suggested by Tans et al. (1990), then the missing sink could have a 1980s-mean value as large as 4 GtC/yr. Such an extreme value is highly unlikely, however, since recent work has supported the IPCC ocean flux range (Sarmiento and Sundquist, 1992; Robertson and Watson, 1992). Using 2.0 +/- 1.0 GtC/yr, as the 1980s-mean ocean flux, yields a missing sink of 1.7 +/- 2.7 GtC/yr*.

The most immediate question arising from this discrepancy is where and how large is the missing sink? A secondary, but equally important issue is: what does the existence of a non-oceanic sink mean for projections of future atmospheric CO₂ increases? More generally, how sensitive are such projections to carbon cycle model uncertainties? These are the questions addressed in this paper.

2. Carbon cycle modelling

Carbon cycle models differ in many aspects, not least in the efficiency with which CO₂ is transferred from the atmosphere to the ocean. Typical box-diffusion models give an ocean flux of 2.0-2.5 GtC/yr for the 1980s (e.g., 2.4 GtC/yr; Siegenthaler, 1983), while ocean general circulation models with embedded carbonate chemistry give values between 1.2 GtC/yr (Maier-Reimer and Hasselmann, cited by Watson et al., 1990, p. l3) and l.9 GtC/yr (Maier-Reimer and Hasselmann, 1987; Sarmiento et al., 1992). The highest values have been obtained with an outcrop-diffusion model, viz 3.6 GtC/yr (Siegenthaler, 1983), although this is generally thought to be an overestimate because the model assumes instantaneous exchange between the atmosphere and the deep ocean through outcrop zones. It is on the basis of such information, together with empirical estimates such as those of Keeling et al. (1989) and Tans et al. (1990) that IPCC suggested the range of 2.0 +/- 1.0 GtC/yr.

When models such as those cited above are used to make projections of future atmosphere CO₂ concentration changes, they are often tuned first to match past concentration changes. This tuning may be performed, not by changing internal model parameters (which may be set on the basis of, for example, isotopic data such as ¹⁴C), but by adjusting the somewhat uncertain history of past net emissions changes to match past concentration changes. This can be done in a straightforward way by inverting the carbon cycle model calculations (e.g., Enting and Pearman, 1987; Siegenthaler and Oeschger, 1987; Wigley, 1991). The implied net emissions clearly depend on the particular model; specifically, on how efficiently the model transfers CO₂ from the atmosphere to the oceans. In the calculations described below, the inversion calculations are generalized by varying the overall magnitude of the ocean flux. The model to be used has a free parameter that can be adjusted so as to match a particular instantaneous ocean flux value (e.g., for a specific year) or integrated flux over a specified time period.

Inverse calculations interpret concentration changes in terms of changes in net emissions, i.e., the sum of industrial emissions (fossil fuel burning, cement production, gas flaring), emissions due to land-use changes (mainly those associated with deforestation), and other unspecified sources and/or sinks. It is the sum of these unspecified sources and/or sinks that corresponds to the missing sink noted earlier. The difference between the net emissions so calculated and independent data for the industrial and land-use-change emissions provides an estimate of changes in the missing sink term.

A number of explanations for the missing sink have been proposed, all of which require the missing carbon to be stored in the terrestrial biosphere. The most obvious possibility is that the land-use-change emissions have been over-estimated, due, for example, to inadequate provision for regrowth (Kauppi et al., 1992) and/or to an underestimation of the amount of burnt material transferred to and stored more or less permanently in the soil zone (Goudriaan, 1989). Alternatively, there may have been an enhancement of biomass productivity due to the CO₂ "fertilization" effect, nitrogen fertilization from industrial pollution, and/or climate change.

If a carbon cycle model is to be used to make future concentration projections, then there are two options; either processes representing the missing sink can be built into the model, or the missing sink can be ignored. If one were to consider only the 1980s budget, then ignoring the missing sink may be justifiable since a zero value lies within the range of uncertainties This is also possible if one considers the integrated budget over 1850-1989 (see Table 1). However, it is not possible to match the full time histories of modelled and empirical net fluxes, even within their wide uncertainty ranges. Even if it were, on the basis of the best estimates of the various components of the carbon budget, it is clearly preferable to build some additional sink mechanism into any model to be used for future CO₂ projections. In other words, the model should in some way, account for the missing sink.

In the present model, only the CO₂ fertilization effect, which acts as a negative feedback effect on concentration build up, is considered. Numerous other models have been developed that include this mechanism (e.g., Bacastow and Keeling, 1973; Björkstrom, 1979; Goudriaan, 1989, Harvey, 1989a). In principle, the strength of the fertilization feedback may be adjusted to optimise the balance of the carbon budget. However, even if this is done, the balance may be spurious if the model ocean produces an incorrect atmosphere-to-ocean flux. Thus, it is an additional advantage if the model allows a range of ocean fluxes to be considered.

3. The carbon cycle model

The model considers six carbon reservoirs: the oceans, the atmosphere, and four terrestrial carbon stores. Terrestrial feedbacks due to changes in temperature and atmospheric carbon dioxide concentration are included (although temperature feedbacks are ignored in the present analysis), and the method by which the atmosphere-to-ocean flux is modelled allows this time-dependent flux to be adjusted in an internally consistent way. Much more complex models exist, but a compromise has to be reached between the detail and transparencyof the model (since the ease with which results may be interpreted tends to be inversely related to the number of potentially tunable model parameters).

The basic equation is the mass balance relationship given by eq. (1), which may be rewritten as

Here, [Delta] C = C - C₀, is the concentration change from a pre-industrial initial level of C₀, E(t) is the total (net) emissions and F(t) the atmosphere-to-ocean flux. E(t) and F(t) are both expressed as changes relative to their pre-industrial levels (it is assumed that steady-state conditions apply initially). In general, F should be the sum of all fluxes into the ocean, including carbon from weathering transported by rivers. The absolute value of the weathering flux is certainly significant (Sarmiento and Sundquist, 1992), but its changes are probably small and they are ignored here.

Emissions may be divided into two components, those due to industrial activity (I), mainly fossil fuel burning and those due to changes in the terrestrial carbon store (viz. dB/dt). The latter in turn may be divided into the net flux due to land-use changes (D_n), mainly deforestation, and carbon store changes associated with the primary terrestrial feedbacks (X). Hence,

so that eq. (2) becomes

The term X represents the missing sink (positive X means a transfer from the atmosphere to the terrestrial biosphere).

To solve eqs. (3) and (4), additional equations determining dB/dt and F are required. dB/dt is determined by the terrestrial component of the model, which is described in detail in the Appendices. The ocean flux, F, is determined here using the convolution integral representation of the ocean general circulation carbon cycle model of Maier-Reimer and Hasselmann (1987); see also Harvey (1989a). As described by Wigley (1991), the model can be written in the form

where [Delta] C_j are partial concentrations defined by

In eq. (6), [alpha] _j and [Tau] _j (j = 0, 4) are convolution parameters (the [alpha] _j sum to one, and [Tau] ₀ = [Infinity]) and the partial concentrations sum to [Delta] C. The partial concentrations do not have to be calculated by direct integration of eq. (6), which is computationally demanding, but can be obtained much more efficiently from the equivalent differential equation

When summed over j, eq. (7) leads to eq. (5). Comparing eqs. (2) and (5) shows that the ocean flux, F(t), is determined by the partial concentrations according to

By a simple scaling of the lifetimes, therefore (i.e., replacing [Tau] _j by [Tau] _j/E, one can alter the ocean flux (see, e.g., Harvey, 1989a, and Wigley, 1991).

Past or future CO₂ concentration changes for any prescribed scenario of industrial emissions, I(t), and land-use change emissions, D_n(t), can now be obtained by solving simultaneously eq. (7), eq. (3) (which defines I(t) in terms of and changes in the terrestrial carbon store, dB/dt), and eqs. (B2), (B3), (B4), (B5) and (B9) (which determine dB/dt as a function of D_n(t), seeAppendix B). In finite difference form, the equations become a simple set of coupled algebraic equations which can be solved accurately and swiftly with a time step of one year. The model contrasts a number of parameters whose values must be set; specifically, the convolution constants [alpha] _j and lifetimes [Tau] _j (seeWigley, 1991, Table 2), various initial terrestrial fluxes and reservoir sizes, constants that determine how fluxes between the terrestrial reservoirs are partitioned, the CO₂ fertilization feedback parameter r or ß, and the ocean flux scaling factor E. The fertilization effect is modelled here using the rectangular hyperbola representation for which the adjustable parameter is r. r is equivalent to the ß factor which appears in the logarithmic formulation. Further details are given in Appendix A. The parameters for and the structure of the terrestrial component of the model are discussed in Appendix B.

4. Explaining past concentration changes

An alternative way to express these discrepancies is to use a carbon cycle model in inverse mode, i.e., to begin with the observed concentration changes and deduce from these the required history of net emissions, E(t). If the model has a deterministic missing sink term, then its time evolution X(t), will be derived automatically by the model. Adding these two terms (E(t) + X(t)) gives the sum of industrial and land-use-change emissions that is required for consistency with the observed concentration changes (see eq. (3)). Thus if I(t) is known, D_n(t) can be obtained. Comparing this D_n(t) with an observationally-based record (such as that of Houghton, 1991) provides another view of uncertainties in the carbon budget.

The derived history of net land-use related emissions, D_n(t), depends, of course, on how the model quantifies the main sink mechanisms, F(t) to the ocean and X(t) to the terrestrial biomass. In the present model, these are determined by the ocean-flux scaling factor, E, and the fertilization factor, r. (The terrestrial sink is also affected by values of the other model parameters in the land component of the model, see Appendix B, but these are taken as fixed quantities here since most of the variability in the results arises through the range of possible values of the fertilization parameter.) Thus, a range of D_n(t) histories can be derived by choosing different values of E and r (within their recognized uncertainty ranges).

Before considering the results of such inverse calculations, the calculation method will so given in a little more detail. The basic problem is to find, for a given value of the fertilization parameter r, the value of D_n,i (the net land-use-change emission in year i) such that the concentration change over the year calculated by the model

is the same as that observed. This is most efficiently done by iteration, required because X_i depends on D_n,i. The number of iterations needed is small provided the first estimate of D_n,i is a good one.

As noted earlier, the resulting D_n(t) depends on the choice of E and r. To characterize the ocean flux term, the mean value of the flux over the 1980s (F(1980s)) can be used instead of E, since, in the convolution representation of the ocean component of the model there is a one-to-one correspondence between E and F(1980s). The range 2.0 +/- 1.0 GtC/yr is used for F(1980s).

For r, an alternative constraint can be applied to define the range of values used. For any choice of values of the pair F(1980s) - r, a unique value of D_n(1980s) is determined. Thus, instead of using r as a primary specification variable, D_n(1980s) may be used. This has the advantage that D_n(1980s) has a recommended (IPCC) uncertainty range, whereas r does not. In performing the inverse calculations, therefore, a particular pair of values for F(1980s) - D_n(1980s) is chosen initially and an additional set of iterations is performed varying r in order to match the chosen value of D_n(1980s). This iterative procedure can also be carried out very efficiently by making a judicious initial choice for r.

The results of such inverse calculations with the present model are shown in Figs. 1, 2 and Table 2. Three sets of values of F(1980s) (1, 0, 2.0 and 3.0 GtC/yr) and D_n(1980s) (0.6, 1.6 and 2.6 GtC/yr) are used. Figs. 1, 2 also show the history of net land-use-change emissions from Houghton (1991, updated).

Fig. 1 shows D_n(t) for the central value of D_n(1980s) and various values of F(1980s). In both Figs. 1 and 2, the early (pre-1860) short-term fluctuations arise from changes in d [Delta] C/dt, while the later fluctuations arise mainly from short-term fluctuations in the history of industrial emissions.

Because [Delta] C(t) changes are smooth, so too are changes in X(t) and F(t), leaving only D_n(t) in the mass balance to compensate for relatively rapid changes in I(t). Clearly, this is just an artefact of the inverse calculation procedure. The detailed changes in D_n(t) must be independent of I(t), so short time scale variations in D_n(t) derived by inverse means have no real significance.

In Fig. 1, it can be seen that the D_n(t) values are relatively insensitive to F(1980s). This is because a higher value of F(1980s) requires a lower r value to maintain the specified value for D_n(1980s), and the consequent reduced terrestrial sink largely compensates for the increased ocean uptake over the whole time period. (For any fixed value of r, the implied D_n(t) history is, of course, quite sensitive to the choice of F(1980s).)

Fig. 2 shows D_n(t) for the central value of F(1980s) and a range of D_n(l980s)values. The general character of D_n(t) shows a rapid increase in the late 18th century, followed by a period of steadily increasing land-use emissions to around 1890. The early rapid increase in D_n(t) is partly an artefact of uncertainties in the observed concentration changes. Relatively minor changes in the concentration data produce noticeably different D_n(t) histories, simply because inverse modelling is very sensitive to dC/dt. For D_n(1980s) = 0.6 GtC/yr, D_n(t) increases to around 1890 and then declines somewhat erratically to near zero in the mid 1970s, before showing the rapid increase that is common to all D_n(1980s) cases. For D_n(1980s) = 1.6 and 2.6 GtC/yr, D_n(t) remains at roughly the same level from 1890 to the mid 1970s. The most striking feature of the recent record is the very rapid rise in D_n(t) in the late 1970s/early 1980s, similar to the Houghton data. The Houghton data, however, begins this rise in 1950, whereas the inverse results do not show a rise until around 1973. Indeed, for D_n(1980s) less than about 2 GtC/yr, the inverse results show a marked decline in land-use emissions over 1945-73, opposite to the Houghton data trend.

Another interesting result from these calculations is the apparent inconsistency between the inverse estimates of D_n(1980s) and the sum of emissions over 1850-1989 ([Sigma] D_n(1850-1989); see Table 2). The IPCC estimates are D_n(1980s) = 1.6 +/- 1.0 GtC/yr and [Sigma] D_n(1850-1989) = 123 +/- 40 GtC (see Table 1). From the Houghton data used here the corresponding values are 1.6 GtC/yr and 121 GtC. For D_n(1980s) = l.6 +/- 1.0 GtC/yr, the inverse calculations give [Sigma] D_n(1850-1989) = 85-200 GtC (if r is constrained to lie in the range 1.0-1.5). Although the two [Sigma] D_n ranges overlap, the inverse calculations suggest a higher value than the observationally-based estimate. Fig. 2 shows that this is due largely to higher inverse deforestation estimates in the earlier part of the record, up to around 1950 for D_n(l980s) = 1.6 GtC/yr, the case which is the most similar to the Houghton data in recent decades.

5. History of the "missing sink"

The results are shown in Figs. 3, 4.

Fig. 3 compares the inverse and Houghton results for different ocean uptake efficiencies, F(1980s) = 1.0, 2.0 and 3.0 GtC/yr. For any given F(1980s) value, the missing sink implied by the inverse calculation is larger than the value implied by Houghton's data up to around 1950. As noted earlier, this is because the inverse values of D_n(t) are greater than Houghton's for all except the most recent decades. The inverse values vary smoothly because they reflect the smoothly varying concentration changes that determine the missing sink in the model. The Houghton values vary on shorter time scales, reflecting similar time scale variations in Houghton's D_n(t) and in the industrial emissions term, I(t). The real-world missing sink should also show decadal and shorter time scale variability, reflecting climate-related variations in terrestrial processes and ocean uptake. The changes evident in Fig. 3, however, cannot be interpreted as such; more likely they reflect uncertainties in D_n(t) and I(t). The near-zero values of the Houghton-based missing sink up to around 1890, the rapidity of the rise over 1890 to 1955 (more rapid than any of the inverse curves), and the levelling off of the sink subsequently should not be over-interpreted. They could as well reflect data input uncertainties as physically meaningful variations in the sink.

In Fig. 3, the implied history of the missing sink was shown to be highly sensitive to the efficiency of ocean uptake. A larger ocean sink obviously implies a smaller missing sink. Fig. 4 shows the sensitivity of the missing sink to uncertainties in the current level of land-use-related emissions for a 1980s ocean flux of 2.0 GtC/yr. In the present model, larger values of D_n(1980s) require a larger CO₂ fertilization effect (i.e. larger r), which in turn leads to a larger missing sink. The Houghton-based missing sink history for F(1980s) = 2.0 GtC/yr is shown for comparison. Since D_n(1980s) for Houghton's data is fixed (and close to 1.6 GtC/yr) it is not possible to estimate D_n(1980s)-related uncertainties for the Houghton-based missing sink.

6. Future concentration changes

Clearly, a model that gives a realistic carbon balance today and in the past must give more defensible future projections than one which does not. Just how important is it to ensure a realistically balanced starting point for projections beyond 1990? At least qualitatively, the answer to this question is obvious. If negative feedbacks are required to balance the carbon budget today, then ignoring these in the future will lead to an over-estimate of future concentration levels. The main argument in favour of the present balanced-model approach is the strong evidence for a CO₂ fertilization effect, both at the small scale (e.g., Gates, 1985; Shugart et al., 1986) and the global scale (e.g., Gifford, 1980; Keeling et al., l989; Enting and Mansbridge, l991).

In the following, future projections will be made under the constraint of a realistic carbon budget balance. This still leaves open the possibility of a range of projections for any given emissions scenario, since there are quite wide uncertainties regarding the balance. Furthermore, any prescribed balance can be achieved in different ways and these may lead to different future concentrations.

Just what is meant by a model that balances the carbon budget? Clearly, any model must give a balance between the rate of increase of concentration and the sources and sinks that the model considers. The balance issue refers to whether the model budget agrees with observations. In other words, we refer somewhat loosely to a "balanced" model as one that leads to no inconsistencies between model output and observations, within recognized and acceptable observational and model uncertainties. Such consistency may be achieved in different ways. For example, if the model were run in forward mode forced by land-use and industrial emissions changes that were within their respective uncertainty ranges, then the implied ocean uptake and atmospheric concentration changes would also have to be within their accepted uncertainty bounds. Alternatively, as done here, if the model is run in inverse mode with given industrial emissions and concentration changes, then the implied land-use-change emissions and ocean uptake histories would have to be within the accepted uncertainty ranges for these quantities.

The procedure used to balance the present model has been described in the previous section. The model is constrained to have 1980s-mean atmosphere-to-ocean and land-use-change fluxes within the IPCC-recommended ranges of 1.0-3.0 GtC/yr and 0.6-2.6 GtC/yr, respectively. For any specified F(1980s) - D_n(1980s) pair, the fertilization feedback parameter r is calculated using the inverse modelling procedure described above. This ensures a perfect match between past modelled and observed CO₂ concentration changes.

For future emissions, the 1992 IPCC scenarios are used (from Leggett et al., 1992). Values were linearly interpolated between those given in Table 3. The results for the central emission scenario (IS92a) are shown in detail in Figs. 5, 6. Fig. 5 shows concentration projections for the IPCC best-guess value of D_n(1980s), 1.6 GtC/yr, and for F(1980s) = 1.0, 2.0 and 3.0 GtC/yr*. As noted earlier, the results are relatively insensitive to F(l980s) because the different r values required to give D_n(1980s) = 1.6 GtC/yr lead to changes in the terrestrial sink that largely offset those in the ocean sink.

In Fig. 6, concentration projections are shown for F(1980s) = 2.0 GtC/yr and different values of D_n(1980s). Fig. 6 also shows a "no-feedback" or "passive biomass" projection for F(1980s) = 2.0 GtC/yr, corresponding to the method used in the 1990 IPCC projections. The differences between the projections in Fig. 6 arise because of different amounts of biomass change. For example, comparing the passive biomass case with the full model result for D_n(1980s) = 1.6 GtC/yr shows a concentration difference of 115 ppmv in 2100 implying an additional sequestering of 244 GtC in the biomass.

For the feedback case with D_n(1980s) = 1.6 GtC/yr, the biomass increases by 264 GtC over 1990-2100, compared with a cumulative net land-use-related loss of 85 GtC. Of this differential of 349 GtC, approximately 7% accumulates in the litter reservoir, 35% in the soil (dead) carbon reservoir and 58% goes into the living biomass. Given that the CO₂ concentration in the year 2100 is well over double the pre-industrial level, enough to cause large changes in plant biomass productivity in small-scale experiments, these amounts seem not unreasonable, although they are, of course, model dependent.

Figs. 7-9 and Table 4 summarize results for the other emissions scenarios. In Fig. 7,the IPCC best-guess value of D_n(1980s) = 1.6 GtC/yr is assumed, while Fig. 8, Fig. 9 show results for D_n(1980s) = 0.6 and 2.6 GtC/yr respectively. The Fig. 7 results differ slightly from those given earlier in Wigley and Raper (1992) for a number of reasons: because of improvements made in the terrestrial component of the model: because the 1980s ocean flux value used here is 2.0 GtC/yr rather than 1.95 GtC/yr used earlier; because 1.6 GtC/yr is used as the best guess value for the 1980s-mean land-use-change emissions (1.1 GtC/yr previously); because the earlier calculations assumed l980s-mean gross land-change fluxes rather than net fluxes as more correctly used here; and because a different functional is now used for the CO₂ fertilization effect. The combined effect of all these factors is to produce lower concentration projections.

Because of the character of the emissions scenarios, and because of lags in the response of the carbon system to emission changes, inter-scenario differences are relatively small out to around 2050. After that the results diverge rapidly and the 2100 concentrations span very wide ranges. Since all of these scenarios are based on the general assumption of "existing policies", these ranges reflect basic uncertainties in predicting future population growth and changes in energy use in the absence of strong controls to reduce CO₂ emissions.

Table 4 gives uncertainty ranges for C(2100), and expresses the 2100 results in terms of the radiative forcing change from 1990 (viz. [Delta] Q = 6.3 ln(C(2100)/C(1990)); see Shine et al., 1990). In terms of radiative forcing changes over 1990-2100, the difference between the lowest and highest emissions scenarios is around 4.3 W/m² (+/- 0.2 W/m^2, depending on the value assumed for F(1980s) and D_n(1980s)). The corresponding forcing range between the low and high concentration projections is 1.0 - 1.4 W/m², depending on scenario. While the inter-scenario uncertainty exceeds that due to the carbon cycle uncertainties considered here by three to four times in the year 2100, the relative importance of these uncertainties is strongly time dependent. Out to around 2020, carbon cycle uncertainties in radiative forcing exceed those due to inter-scenario differences.

7. Conclusions

When the carbon cycle model is run in inverse mode, even given its flexibility with regard to the choice of the efficiency of ocean CO₂ uptake (through F(1980s)) and land-use-change fluxes through D_n(1980s)), the implied history of past land use emissions up to 1950 differs markedly from the observationally-based record of Houghton (1991) with the latter being substantially smaller. For D_n(1980s) below about 2.0 GtC/ yr, the modelled D_n(t) series declines noticeably over 1950-75, in contrast to the Houghton data. After 1975, both modelled and observationally-based land-use emissions increase rapidly to the mid 1980s, declining slightly or leveling off thereafter.

A further manifestation of these results is that modelled values of [Sigma] D_n(1850-1989) appear to be inconsistent with the D_n(1980s) values when compared to Houghton's estimates of these quantities. The differences between the inverse results and Houghton's data could reflect either model deficiencies, and/or uncertainties in the Houghton data, the input concentrations or industrial emissions. The neglect of natural processes (such as climate variations) that could alter terrestrial emissions or ocean uptake may account for some of the decadal time scale differences, but not the consistent longer-term discrepancies.

After balancing the contemporary carbon budget (i.e., ensuring that the model's budget breakdown over the 1980s is consistent with observational evidence), projections of future concentration changes were made for the 1992 IPCC emissions scenarios. A range of projections was obtained for each scenario by considering different values for F(1980s) and D_n(1980s), within their current uncertainty ranges. This is not the first time that some form of budget balancing has been achieved by model tuning prior to making CO₂ concentration projections (see, e.g., Bacastow and Keeling, 1973; Harvey, 1989a). However, it is the first time that inverse calculations spanning a wide range of ocean and land-use-change flux uncertainties, and matching past concentration changes precisely, have been carried out as a precursor to deriving a comprehensive range of future concentration possibilities.

The concentration projections for any given emissions scenario show a wide range of uncertainty. While these uncertainties are related indirectly to model parameter uncertainties, they arise here through current uncertainties in the carbon balance, as epitomized by the magnitude of the missing sink. The primary control on the range of uncertainly is D_n(1980s), the emissions due to land-use changes. This sensitivity of projections to the assumed magnitude of past land-use-change emissions has been noted previously by Harvey (1989a). The more general dependence on the magnitude of the missing sink highlights the importance of obtaining a better explanation for the processes that explain and control this sink.

Concentration projections must depend, not only on the current and past magnitude of the missing sink, but also on its future changes. By accounting for the sink using CO₂ fertilization alone, the present model constrains the form of these future changes. If other processes are important, these may change differently in the future.

Nevertheless, the range of projections produced corresponds to a wide range of changes in the magnitude of the sink with time.

In the high concentration projections, the integrated amount of carbon sequestered by the missing sink over 1990-2100 is 108-149 GtC (the value depends on the scenario). The high projections correspond to a missing sink of 0.68 GtC/yr averaged over the 1980s (see Fig. 4) equivalent to 75 GtC over 110 years. These projections therefore correspond to a slight to moderate increase in the magnitude of the sink over the next century. At the other extreme, the low concentration projections correspond to sequestration of 370-705 GtC over 1990-2100 (although it should be noted that these results correspond to an unlikely high value of the fertilization effect; see Table 2). The mean sink strength over the 1980s for these projections is 2.68 GtC/yr (i.e., 295 GtC over 110 years), so the proportional increase in the magnitude of the missing sink is similar to that for the high concentration projections. These results are consistent with the substantial concentration increases over 1990-2100 and the assumption that CO₂ fertilization is the primary explanation for the missing sink.

In conclusion, the simplicity of the model used here must be stressed. Potentially important processes, such as changes in terrestrial fluxes associated with both past and future changes in climate, have not been considered. These effects could either reduce (Harvey, l989b; Prentice and Fung, 1990; Smith et al., 1992) or increase (Oechel et al., 1993; Smith and Shugart, 1993) future concentrations over those calculated here. They represent a significant additional source of uncertainly beyond that considered here.

Large uncertainties remain in carbon cycle modelling, and these can only be reduced by more detailed modelling studies supported by observational and process-oriented research. Of prime importance, however, is the need to reduce uncertainties in all aspects of the land-use component of CO₂ emissions.

8. Acknowledgments

9. Appendix A

The two most commonly used forms for the CO₂ fertilization effect are the logarithmic form introduced by Keeling and Bacastow (1973) and the rectangular hyperbolic or Michaelis-Menton form (Gates, 1985). The former expresses the fertilization enhancement of productivity in terms of a single "ß factor" via the expression

where N is net primary productivity, C is concentration and subscript zero denotes an initial or reference value. Eq. (A1) behaves unrealistically at both low and high concentrations. In particular, it allows NPP to increase without limit as C increases. The hyperbolic form is

Eq. (A2) behaves more realistically in that it leads to zero NPP at C= C_b and has a limiting value as C tends to infinity of

I employ eq. (A2), as given by Gifford (1993), and take N₀ and C₀ to correspond to pre-industrial conditions. Following Gifford, I use C_b = 31 ppmv; other authors have used higher values, but the results presented in this paper are insensitive to the precise value of C_b that is used. To determine the other parameter in eq. (A2), b, Gifford uses the enhancement in NPP that occurs for a PCO₂ increase from 340 ppmv to 680 ppmv

He suggests that a realistic range of values for r is 1.1 - 1.4. From eq. (A2), r is related to b by

Specifying r, therefore, determines N/N₀ for given C in the same way that N/N₀ is determined by ß in the logarithmic form.

The main distinction between eqs. (A1) and (A2) lies in their behavior for large C (> 1000 ppmv). For any given r value, a ß value can be chosen so that the two formulae behave quite similarly over a wide range of concentrations. Nevertheless when the ß and/or r values are obtained by tuning the model to past data, the two fertilization formulations give noticeably different results even for concentrations substantially less than 1000 ppmv.

A convenient way to compare the two formulations is through an equivalent ß value (ß*), defined by

Clearly, ß* is constant (independent of C) for the logarithmic formula. For the hyperbolic formula

which tends to zero for large C. For larger and larger C, the hyperbolic formula therefore gives smaller and smaller enhancement of NPP relative to the logarithmic formula. In describing the results of inverse calculations in which r is adjusted to obtain in a specified value for the mean net deforestation flux over the 1980s to facilitate comparison with the standard ß factor of the logarithmic formulation, I give both the r value, and the value of ß* at 340 ppmv calculated using eq. (A7) and taking C₀ to be the pre-industrial concentration (~278 ppmv).

10. Appendix B

The terrestrial component of the carbon cycle model is essentially one of the hierarchy of globally-integrated box models described by Harvey (1989a). It is a 4-box model, with two living biomass boxes (where living biomass comprises woody material, leaves/needles, grass, roots, etc.) and two dead biomass boxes. One of the living biomass boxes is a rapid turnover box, assumed to have a zero time constant (left box in Fig. 10). A fraction of gross primary productivity (G) cycles through this box directly back into the atmosphere and may be ignored on time scales of interest here. The remaining part of the gross primary productivity is partitioned through the rapid turnover box into fractions which enter the other living biomass box (mass of box P, flux into box G_P), a detritus box (mass H, flux G_H) and a soil box (mass S, flux G_S). The mass-P box (or plant box) has G_P as its sole source term; its sinks are respiration (R), litter production (L: L_H to the detritus box and L_S to the soil box) and a land-use-change component (D_P). The detritus box has sources from gross primary productivity (G_H) and litter production (L_H); its sinks are fluxes to the atmosphere (Q_A), the soil box (Q_A) and land-use changes (D_n). The soil box has sources from the detritus box (Q_S), litter production (L_S) and gross primary productivity (G_S); its sinks are through oxidation to the atmosphere (U) and land-use changes (D_S).

The equations describing mass balances in each box and in the atmosphere (mass M) are (cf. eqs. (2) and (3)):

where D_g = D_P + D_U + D_S, F is the flux into the oceans (i.e., dO/dt where O is the oceanic carbon mass) and I is the input from industrial emissions,

where Q = Q_A + Q_S, and

Note that D_g is gross deforestation, related to net deforestation by

The sinks L, Q and U are assumed to be proportional to the P, H and S box masses respectively, via appropriate time constants;

For the plant box, the term L = P/[Tau]_P ensures that the total plant mass will relax back to its initial state if perturbed by a pulse land-use change (in the absence of changes in G_P and R). This relaxation acts as an effective regrowth term, so that D_P can only be interpreted as a component of gross land-use emissions. Gross and net land-use-change emissions are related by regrowth, W. Following Enting and Lassey (1993), this is determined by

The fertilization effect is incorporated using the rectangular hyperbola expression for changes in NPP, as described in Appendix A. While this strictly applies only to NPP, the model requires separation of the effects on GPP and respiration. For simplicity it is assumed that

where N/N₀ is given in AppendixA.

The model parameters are now as follows: initial values are required for the plant, detritus and soil reservoirs (P₀, L₀ and S₀) and for GPP and respiration (G₀ and R₀); and partitioning factors are required for the flux out of the plant box ([Phi] P is assumed to go to the detritus (1- [Phi]) P to the soil), the flux out of the detritus ([lambda] L is assumed to go to the soil (1 - [lambda]) L to the atmosphere), the division of GPP (g_PG to the plant box, g_LG to the litter and (1- g_F - g_T) G to the soil), and the division of land-use-change emissions (D_g = D_H + D_L +D_S = [Gamma]_PD_g + [Gamma]_LD_g + (1 - [Gamma]_P - [Gamma]_L) D_g. Since the time step used for the model is one year, g_LG represents the annual accumulation of GPP in the litter due to dying above-ground plant matter, while (1 - g_P - g_I) G is the annual accumulation of GPP in the soil due to root turnover. Based on estimates in the literature (see, e.g., Harvey, 1989a) the following parameter values are used: P₀ = 750 GtC, L₀ = 80 GtC, S₀ = 1450 GtC, G0 = 76 GtC/yr, R₀ = 14 GtC/yr, [Phi] = 0.98, [Lambda] = 0.05, g_P = 0.35, g_L = 0.60, [Gamma]_P = 0.70, [Gamma]_L = 0.05. Results are relatively insensitive to the precise choice of parameter values.

The time constants, [Tau]_P, [Tau]_L, and [Tau]_S are determined fully by the other parameters and may be obtained directly from the initial (steady-state) forms of eqs. (B2), (B3), and (B4). The above parameter values lead to [Tau]_P ~ 60 yr, [Tau]_H ~ 1.4 yr and [Tau]_S ~ 209 yr.

* Current address: Office for Interdisciplinary Earth Studies, University Corporation for Atmospheric Research, Boulder, CO 80307-3000, USA.

* Note that if the uncertainty ranges for the individual carbon budget components represent p% confidence intervals, then their sum, +/- 2.7 GtC/yr, will be the P% interval where P>p. For comparability, the equivalent p% range is around +/- 1.5 GtC/yr (see also Table 1).

* There is a minor inconsistency between the choice of D_n(1980s) used to calculate r and the scenario value used for D_n(t) in 1990 (see Table 3). For D_n(1980s) = 1.6 GtC/yr, the model gives D_n(1990) ~ 1.6 GtC/yr, noticeably higher than the IPCC scenario value of 1.3 GtC/yr. In the calculations, D_n(t) was assumed to relax from the 1990 value determined by the inverse calculation to the scenario value over the period 1990-2000.

REFERENCES

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