Interannual Variability

It is possible to perform a similar analysis of the interannual variability of ocean flux (and its relationship to atmospheric concentration measurements) as for the long-term mean. Once again, one cannot consider the ocean alone when interpreting atmospheric signals, so I will consider the combined effect of various scenarios of land and ocean fluxes in the atmosphere. As with the long-term means, even the broadest question regarding interannual variability in atmospheric CO, growth-rate is unsolved, namely, whether the variability is driven predominantly by the land or the ocean. In general, modelers and measurers of fluxes in both environments would suggest that the terrestrial biosphere is largely responsible, with perhaps three or four times the interannual variability of the ocean. Examples from two calculations by Friedlingstein et al. (1997) and Le Quere et al. (2000) are shown as the dotted lines in Fig. 3. Also shown as solid lines are the equivalent fluxes from Rayner et al. (1999) while the asterisks indicate the results of Francey et al. (1995). The divergence is clear, particularly for the ocean, with Francey et al. (1995) showing much greater variability than Le Quere et al. (2000). Rayner et al. (1999) has less variability than Francey et al. (1995), particularly through the 1980s, but still greater than that suggested by the ocean model. The disagreement between Rayner et al. (1999) and Francey et al. (1995) is perhaps surprising since they use the same 8,3C record from Cape Grim. Rayner et al. (1999) uses a global network of C02 observations and a constraint toward an invariant prior estimate, both of which may reduce the estimated variability.

There is little apparent agreement on the long-term mean land flux. As already mentioned, this disagreement is mainly because of a difference in the quantities plotted; Rayner et al. (1999) and Francey et al. (1995) plot total terrestrial flux while Friedlingstein et al. (1997) omit the flux due to land-use change. Once this is taken into account the mean estimates of Friedlingstein et al. (1997) and Rayner et al. (1999) are consistent. There is also more agreement for the three calculations on the magnitude of interannual variability. There is agreement on the timing of some events, like the large anomalous sources of 1987 and subsequent decrease, but I do not regard the estimates as in overall agreement.

In an atmospheric inversion, the atmosphere acts as a consistency check between land and ocean estimates since their sum must match the growth rate and spatial gradient information in the concentration observations. This can be implemented by using the flux estimates as a prior constraint, with confidence dictated by the providers of the estimates. While the calculations here have not done this, experience shows that the estimated fluxes would be little different unless I used

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FIGURE 3 12-month running mean fluxes to the atmosphere (Gt C year-') for the ocean (a) and land (b) from the inversions of Rayner et al.

(1999) and Francey et ill. (1995) and the flux models of Le Quere et al.

(2000) for the ocean and Friedlingstein et al. (1997) for the land.

unrealistically tight prior constraints. Hence, the flux estimates are inconsistent with atmospheric observations as used here.

There are many potential explanations for the inconsistency. The first is that the inversion misallocates variability between the land and ocean. This is certainly possible since the observing network is strongly biased toward the ocean. It is part of the behavior of such Bayesian inversions that they will adjust those fluxes that are best sampled to make up for a mismatch between data and the initial guess. Thus the inversion will propagate interannual variability in concentration data preferentially to the relatively well-sampled ocean rather than the poorly sampled land. This effect is reduced by the use of <5BC, a tracer that marks terrestrial but not oceanic net sources. Both the Rayner et al. (1999) and Francey et al. (1995) calculations use SI3C. In fact a similar calculation by Keeling et al. (1995) had such large interannual variability in SBC that the deduced terrestrial fluxes required large compensating oceanic interannual variability to match the global C02 growth rate. Note that at the global scale, potential errors in the modeling of atmospheric transport (which would manifest themselves as misallocation of sources) do not matter. Further, there is broad agreement on the dominant role of the tropics, both land and ocean, in forcing interannual variability so the transport seems to be behaving consistently.

Another possibility is that the flux estimates are incorrect. For example, the ocean model of Le Quere et al. (2000) is relaxed toward climatology outside the tropics so that interannual variability is suppressed. However, long experience of the oceanographic community has identified the El Nino Southern Oscillation (ENSO) as the dominant oscillation on these time-scales. Further, the magnitude of variations in the ocean model in tropical fluxes is roughly supported by measurement campaigns of Feely et al. (1999). There are also processes missing from the terrestrial estimates, e.g., interannual variations in disturbance or changes in surface solar radiation. It is an unfortunate consequence of the integrated nature of the atmospheric constraint that while it can project information from one region to another, it also projects errors.

Several ways forward through this paradox are apparent, some by analogy with the approach to the long-term mean problem, but some suggested by the nature of the interannual variability problem itself. First, an obvious need is for more data, subject to concerns about model error. In particular, there is need for more concentration data over the tropical continents. Such data would reduce the leakage problem mentioned above in which concentration variations observed at marine sites (but possibly forced by remote continental fluxes) are attributed to the ocean. The next need is for continuing reanalysis of the relative confidence in ocean and terrestrial flux estimates. The atmospheric constraint allows land and ocean flux estimates to inform each other, but only on the basis of credible uncertainty estimates. In the study of Rayner et al. (1999) uncertainties were chosen almost arbitrarily, the main requirement being to avoid the risk of bias from too tight a prior constraint. Only the workers constructing the flux estimates themselves, either through an understanding of the scaling properties of observed fluxes, or through model sensitivity, can provide more credible estimates. This must happen before the scientific community can be sure we are focusing on real disagreement rather than being misled by an overoptimistic assessment from inverse modelers.

The final and most radical suggestion concerns a different way of modeling the problem. Partly the suggestion comes from reconsidering the question of why the carbon-cycle community is interested in interannual variability. While the task of estimating the interannual variability of fluxes is difficult enough to have become an end in itself, it is not sufficient. The next step is to elucidate those processes that control the variations in flux. This is a fascinating scientific question but also has practical import since it should enable a

Zwierz Namalowane Kropek

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FIGURE 3 12-month running mean fluxes to the atmosphere (Gt C year-') for the ocean (a) and land (b) from the inversions of Rayner et al.

(1999) and Francey et ill. (1995) and the flux models of Le Quere et al.

(2000) for the ocean and Friedlingstein et al. (1997) for the land.

us to estimate the sensitivity of the global carbon cycle to various forcings. The processes that control the variations are, or should be, expressed in models of the processes with the models forced by various boundary conditions. The unknown parameters of the problem now move from flux estimates themselves to the various model parameters and, potentially, boundary conditions of the underlying process models. The problem now becomes a nonlinear optimization problem but apart from this technical issue, little else changes from current flux inversion methods.

Such an approach has many advantages. First, it allows the atmosphere to inform directly the understanding of the relevant processes. Second, the approach addresses one of the great difficulties in an integrated approach to understanding the carbon cycle, the difference in the scales on which parameters are estimated or measured. Generally, direct flux measurements or campaigns estimate fluxes on a very small scale or (better) allow estimates of some parameters controlling, say, the terrestrial biosphere. Rightly are the methods termed bottom-up. They return quantities that are hard to compare with the large-scale estimates returned from synthesis inversions. If the methods were used to estimate the same parameters in some underlying model we could make a direct comparison. Finally, there are computational attractions. The very enterprise of modeling fluxes (from process models) assumes that variations in flux can be computed from some relatively well-known set of forcings and some set of model parameters. This set is almost certainly smaller than the set of fluxes now estimated by a flux-based inversion. We will see below that this could help overcome some of the restrictions on resolution that have plagued this type of study.

To see how such an approach may work, recall that the current inversion methods involve an optimization problem, seeking to minimize a cost function, comprising a mismatch between modeled concentrations (D) and observed data (Do) plus another term for the mismatch between estimated sources (S) and an initial estimate of those sources (S„), cf = 0 - D0)rC~'(D0)(D - D0) + (S ~ S0)rC-l(Sa)(S - S„) (2)

where C21(X), an inverse covariance matrix, expresses the confidence in a quantity X.

Sources and data are related by the linear operator J as

where J embodies the transport model. Viewed more generally, J is the sensitivity of data with respect to sources V^ D .

Now, replace S with the output of some process model, M,as

where P represents model parameters. The task is to estimate P given D and perhaps an initial estimate P0. The estimation usually requires V^ D

to aid minimization of the cost function. We can invoke the chain rule to expand the derivative as

The first term in the product is the previous J derived from the transport model. The second term has previously been tedious to calculate in an optimization routine since it requires a numerical coding of the derivative of the process model. The emergence of automatic differentiation tools (e.g., Giering, 2000) will greatly facilitate the approach.

The dimension of S no longer appears in the optimization problem. So, provided the procedure has access to J at high resolution, it is possible to avoid some of the problems of aggregating fluxes into large regions. Using automatic differentiation techniques, Kaminski et al. (1999) calculated J for a full transportmodel grid (8° X 10°) and a network of a few dozen observation sites. At this resolution, over 20 years, we would need to solve for approximately 200,000 flux components, while the parameter approach may use only hundreds of unknowns.

Such an approach is slightly easier over land than over the ocean since the process-models are single-point models, so the derivatives are easier to calculate. An example of using atmospheric observations this way (although without the formal inversion procedure) can be found in the study of Fung et al. (1987). They estimated the seasonal cycle of net biospheric CO, flux to the atmosphere from satellite and field data and used the generated seasonal cycle of C02 to test some of the details of their formulation. A more recent example is a single-column model of ocean biology in the study of Balkanski et al. (1999). These workers used atmospheric oxygen data and some assumptions about the contribution of the terrestrial biosphere to the seasonal cycle to calculate gross and export production in a simple column model (replicated over the ocean surface). They also did not perform a formal inversion but they did adjust model parameters until they achieved a near-optimal match. Historically, this is the path by which the flux-based inversions proceeded too. Ad hoc adjustment of fluxes was used by, e.g., Tans et al. (1990) for carbon dioxide or Fung et al. (1991) for methane and only later replaced by optimization algorithms by Enting et al. (1995).

The approach outlined above would make substantial demands on both the atmospheric inversion community and process modelers. It would shift the boundaries of the atmospheric inversion task from providing estimates against which process models can be tested to formally integrate those models into the procedure. It may also use the atmospheric constraint as just one among many acting to constrain model parameters, since any observable derived from P can be used. The approach would also make serious demands on process modelers. If they are to use the atmospheric constraint, the models must be comprehensive models of flux to the atmosphere. A process model that attempts to estimate, say, only net primary productivity cannot simulate the flux to the atmosphere and therefore cannot be used this way. Despite its difficulties, the approach seems to offer an integrated and rigorous framework for the use of atmospheric data in studying the carbon cycle.

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