I. G. Enting

Inverse problems in atmospheric constituent transport

Part I. Principles: 1. Introduction 2. Atmospheric transport and transport models 3. Estimation 4. Time series estimation 5. Observations of atmospheric composition 6. The sources and sinks 7. Problem formulation 8. Ill-conditioning 9. Analysis of model error 10. Greens functions and synthesis inversion 11. Time-stepping inversions 12. Non-linear inversion techniques 13. Experimental design Part II. Recent Applications: 14. Global carbon dioxide 15. Global methane 16. Halocarbons and other global-scale studies 17. Regional inversions 18. Constraining atmospheric transport 19. Conclusions References Appendix A. Notation Appendix B. Numerical data Appendix C. Abbreviations and acronyms Appendix D. Glossary Appendix E. Data source acknowledgements Problems.

On aggregation errors in atmospheric transport inversions

This paper explores the consequences of resolution of surface fluxes on synthesis inversions of carbon dioxide. Synthesis inversion divides the Earths surface into a set of regions and solves for the magnitudes of fluxes from these regions. The regions are generally quite large. By considering an inversion performed at the resolution of the underlying transport model we show that the aggregation to large regions can cause significant differences in the final results, with errors of the same order of magnitude as the fluxes themselves. Using a simple model, we derive an algorithm to reduce this error. This algorithm accounts for the extra data uncertainty that is caused by uncertainty in the small-scale flux components. In the spatial synthesis inversion this extra data uncertainty reaches a maximum value of 3.5 ppmv. Accounting for it can halve the aggregation error. We provide suggestions for dealing with this problem when high-resolution inversions are not feasible.

Modeling air movement and bubble trapping in firn

A finite difference model for gas diffusion and bubble trapping in firn is described. The model uses prescribed profiles of density, open and closed porosity, and diffusivity to determine the diffusion and trapping processes. The model is calibrated and tested by using measured air composition in the firn at the DE08-2 site on Law Dome, Antarctica. In particular, we focus on carbon dioxide (CO2), methane (CH4), and sulfur hexafluoride (SF6), which have well-determined atmospheric records (CO2 since 1958, CH4 since 1983, and SF6 since 1978). These trace gases are used to tune the diffusivity-porosity relationship, which is the most uncertain of the model inputs. Modeled trace gas profiles in the DE08-2 firn are improved if allowance is made for reduced diffusion through the most prominent DE08-2 melt layer from the summer of 1989/1990. The relatively rapid growth rate of SF6 in the atmosphere permits good definition of the diffusion reduction due to the melt layer (about 80%). The model quantifies the smoothing effect of the firn diffusion and bubble trapping on atmospheric signals. Gravitational separation in the firn is investigated by comparison of modeled δ15N2 with observations. The model is used to calculate the isotopic diffusion correction for δ13C02 and δ13CH4. This corrects for the fractionating effects of the firn diffusion process on the different isotopes. The diffusion and gravitational corrections are critical at the measurement precision currently being obtained; for δ13CO2 the diffusion correction is up to about 10 times the current measurement precision. The diffusion correction is even more significant for δ13CH4; at over 1‰ at the bottom of the firn it is more than double the change over the last decade. The fully corrected δ13C02 record from the DE08-2 firn is compared with the history of Cape Grim direct atmospheric measurements with excellent agreement.

Seasonal and latitudinal variability of troposphere Δ14CO2: Post bomb contributions from fossil fuels, oceans, the stratosphere, and the terrestrial biosphere

initially caused by stratosphere-troposphere exchange, with the injection of bomb 14 Ci nto the troposphere during winter and spring mixing. Here we show how fossil, ocean, and terrestrial biosphere fluxes modified the stratospheric signal during the 1960s, and the evolution of each of these components in the post bomb era. In our analysis, we used the Goddard Institute for Space Studies (GISS) atmospheric tracer model, gross ocean CO2 fluxes from the Lawrence Livermore National Laboratory (LLNL) ocean model, and terrestrial CO2 fluxes from a biosphere-atmosphere model driven by normalized difference vegetation index and surface air temperatures. We found that 14 C-depeleted respiration from the terrestrial biosphere partially canceled the 14 C-enriched stratosphere flux in the Northern Hemisphere in the mid and late 1960s. In more recent decades, our analysis suggested that the terrestrial biosphere contribution to the D 14 C seasonal cycle reversed phase, with the terrestrial biosphere currently releasing relatively 14 C-enriched CO2 that mixes with relatively depleted troposphere CO2. The timing of this reversal depended on the residence times of carbon within the footprint of the observation station. Measurements of D 14 C in respiration from tundra and boreal ecosystems in Alaska provide evidence that some boreal forests have undergone this transition, while some tundra ecosystems have not. We predict that over the next century, several features of the latitudinal profile of D 14 C will substantially change because of continued fossil fuel emissions in the Northern Hemisphere, and the partial release of bomb 14 C that has accumulated in Southern Hemisphere oceans. INDEX TERMS: 0368 Atmospheric Composition and Structure: Troposphere—constituent transport and chemistry; 1610 Global Change: Atmosphere (0315, 0325); 1615 Global Change: Biogeochemical processes (4805); 3339 Meteorology and Atmospheric Dynamics: Ocean/atmosphere interactions (0312, 4504); KEYWORDS: radiocarbon, Suess Effect, carbon cycle, Arctic and boreal ecosystems, Amazon tree rings, meridional gradient of 14 C

Algebraic techniques for enumerating self-avoiding walks on the square lattice

The authors describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3N/4 times a polynomial in N, and so the approach is greatly superior to direct counting techniques. They have enumerated walks of up to 39 steps. As a consequence, they are able to accurately estimate the critical point, critical exponent, and critical amplitude.

Data and modelling requirements for CO2 inversions using high-frequency data

We explore the future possibilities for CO2 source estimation from atmospheric concentration data by performing synthetic data experiments. Synthetic data are used to test seasonal CO2 inversions using high-frequency data. Monthly CO2 sources over the Australian region are calculated for inversions with data at 4-hourly frequency and averaged over 1 d, 2.5 d, 5 d, 12.17 d and 1 month. The inversion quality, as determined by bias and uncertainty, is degraded when averaging over longer periods. This shows the value of the strong but relatively short-lived signals present in high-frequency records that are removed in averaged and particularly filtered records. Sensitivity tests are performed in which the synthetic data are ‘corrupted’ to simulate systematic measurement errors such as intercalibration differences or to simulate transport modelling errors. The inversion is also used to estimate the effect of calibration offsets between sites. We find that at short data-averaging periods the inversion is reasonably robust to measurement-type errors. For transport-type errors, the best results are achieved for synoptic (2–5 d) timescales. Overall the tests indicate that improved source estimates should be possible by incorporating continuous measurements into CO2 inversions.

Inverse problems in atmospheric constituent studies: III. Estimating errors in surface sources

For pt.II, see ibid, vol.6, p.349-62 (1990). A simplified semi-analytic model of atmospheric transport is used to obtain error estimates for sources deduced from the spatial distribution of greenhouse gases. These estimates quantify the extent of the trade off between resolution and variance in the inversion problem. They show that the results of atmospheric transport modelling cannot be readily incorporated into studies yielding regional AUK estimates for trace gases, without taking explicit account of the reduced resolution arising in the inversion of concentration data.

The number of convex polygons on the square and honeycomb lattices

A subset of the set of self-avoiding polygons (SAP) embeddable on the square lattice which display the property of convexity is defined. An algorithm for their enumeration is developed, and from the available series coefficients the exact generating function is found. The singularity structure appears similar to that of the unsolved SAP problem, but with different critical exponents and critical points. The enumeration of convex polygons allows the extension of the existing series for the square lattice SAP by one term. For the honeycomb lattice similar results have been obtained, despite a less natural definition of convexity.

Fire spread and percolation modelling

Simple percolation theory, which has recently been used as a mathematical model of fire propagation, predicts values for critical exponents that differ from those obtained by a laboratory simulation of percolation in which matchsticks (with ignitable heads) are randomly placed in a square lattice. The most noticeable discrepancy is that the theory predicts that at critical percolation a fire-front decelerates, whereas the experiments indicate acceleration. Although site-bond percolation theory yields qualitative insights into expected fire-behaviour, its present formulation appears to be quantitatively incorrect. This implies that actual fires are in a different universality class to site-bond percolation.

The size and number of rings on the square lattice

The authors have calculated the number of self-avoiding polygons on the square lattice to 56 steps, and the caliper size to 54 steps. Analysis of the generating function permits estimates of the connective constant, mu =2.638 1585+or-10-6 and the critical exponents alpha =0.500 06+or-0.000 06 and v=0.753+or-0.007. The singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent Delta =1.5, as predicted by Nienhuis (1982, 1984). The confluent part, however, maps into the additive analytic term due to the value of the exponent alpha .