SELF-ORGANIZED CRITICALITY IN CLOSED ECOSYSTEMS: CARBON DIOXIDE FLUCTUATIONS IN BIOSPHERE 2

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1 NTERNATIONAL JOURNAL OF CLIMATOLOGY, VOL. 16, (1 996) SELF-ORGANIZED CRITICALITY IN CLOSED ECOSYSTEMS: CARBON DIOXIDE FLUCTUATIONS IN BIOSPHERE 2 RAYMOND J. CRONISE, DAVID A. NOEVER AND ANDREW BRlTTAIN Biophysics Branch. ES76, NASA Marshall Spaceflight Center Huntsville, AL 35812, USA Received 23 December I994 Accepted I5 September 1995 ABSTRACT A little understood question in climate and ecological modelling is when a system appropriately can be considered in statistical equilibrium or quasi-steady state. The answer bears on a host of central issues, including the ability of small perturbations to cause large catastrophes, the constant drift of unsettled systems, and the maximum amount of environmental control theoretically possible. Using Biosphere 2 records, the behaviour of carbon dioxide fluctuations was tested for correspondence with theories now known collectively as self-organized criticality. The signature of agreement with other large, composite systems, including forest tires, stock markets, and earthquakes, is a common frequency spectrum or power-law correlations. In this case, the largeand small-scale ends of the spectrum share a common driving force and consequently no single cut-off exists for excluding or ignoring small environmental changes. From the Biosphere 2 carbon dioxide data, the fluctuations in internal atmospheres vary in both small and large steps. The time fluctuations were examined as they varied over 2 years and over three orders of magnitude in fluctuation size, then binned into characteristic size classes. The statistics show a power-law scaling exponent of - 1.3, compared with - 1 for classical flicker noise (l/fspectrum) and -2.5 for analogous sand-pile experiments developed to test the predictions of a self-organized, critical system. For comparison with open ecosystems, the Byrd climatic record of global COz over the last 50 ka has a similar power-law relation but with -2.3 as the scaling exponent. For generalizing self-organized criticality, the design suggests that otherwise unrelated biological and physical models may share a common correlation between the frequency of small and large length-scales or equivalently exhibit temporal similarity laws. The results potentially have wide implications for environmental control in otherwise chaotic or difficult to predict ecological behaviour. KEY WORDS: carbon dioxide fluctuations; self-organized criticality; Biosphere 2. As introduced by Bak et al. (1988a,b), the notion of self-organized criticality (SOC) supposes that many-bodied physical systems first assemble into a critical state, then relax from small perturbations via both minor and catastrophic changes. One model system is a sand-pile near its critical angle of repose (Kadanoff et al., 1989; Held et al., 1990; Diodati et al., 1991; Evesque, 1991). As single grains of sand are added, the size and frequency of avalanches remain correlated. This correlation can be taken as a general quality of some large, composite systems including earthquakes, stock markets, forest fires, weather, and ecology (Bak and Chen, 1991; Babcock and Westervelt, 1989; Schwartz, 1990). Its primary observable is a power-law scaling between either the size or duration of an avalanche event and its frequency. Variations in behaviour at small and large scales, although occurring with different frequencies, arise independent of the path taken by the system in its approach to its critical state and thus deserve consideration as a universal property of many-body statistics. The present aim is to examine the relevance of these statistical models to explain atmospheric fluctuations in closed ecosystems. The Biosphere 2 system is a closed ecological facility including a range of biome classes: rainforest, savannah, thomscrub, desert, fresh- and saltwater marsh, beach, and coral reef ocean. Its species diversity at closure totalled 3000 plant and animal types. Because environmental control technologies maintained climate, air movement, streamflows, and waves, human management of carbon dioxide made up a critical component of the biosphere. Fluctuations in atmospheric carbon dioxide are less amenable to a non-statistical analysis, owing principally to the cutting and storing of plant biomass, manipulation of night-time temperatures, and operation (Roenneberg et al., 1989) of a recycling system to chemically remove carbon dioxide in the form of CaC03. For this reason, the closed ecosystem differs from many previous laboratory settings (Noever, a,b) in its diversity and range of inputs, both anthropogenic and natural. CCC /96/ by the Royal Meteorological Society

2 598 R. J. CRONISE, D. A. NOEVER AND A. BRlTTAIN Compared with a 3-year lifetime in the Earth s atmosphere, the residence time of carbon dioxide in Biosphere 2 was less than 10 days (Brittain, 1993). This more rapid turnover gave rise to larger fluctuations, ranging from a low of 1000 ppm to over 4000 ppm. Clouds in particular served as a sensitive external driver for Biosphere 2 s carbon dioxide, as did light. Diurnal flux reached up to ppm. These changes counterbalanced a consistent drop in atmospheric oxygen (6.1 per cent) attributable to oxidation of soil organic matter to produce carbon dioxide and subsequent reaction with calcium in the structural concrete to form calcium carbonate. Carbon isotope analysis and mass balance relations confirmed this sink for atmospheric oxygen and occasioned an oxygen injection as levels fell below 17 per cent (Brittain, 1993). This experimental observation in Biosphere 2 likewise confirms previous work (Kearns, 1983; Obenhuber, 1986; Schaffer, 1991; Brittain, 1993) with replicate microbial closed ecosystems, which has shown (Brittain, 1993) the independence of gas phase oxygen concentration to overall carbon redox balance. The question of whether Biosphere 2 s atmosphere developed into a self-organized critical state centres on the spectrum of its carbon dioxide fluctuations. Although a characteristic negative power law indicates that small atmospheric changes occur with higher frequency, the same physical mechanism should govern both large- and small-scale action. For comparison with existing SOC models, we shall refer to atmospheric fluctuations as avalanches or events that can trigger changes of all sizes. One result unique to SOC is that even minor fluctuations can drive large-scale changes or catastrophes. This corresponds qualitatively to the so-called straw that broke the camel s back, although the critical state recovers its repose without destroying its basic physics; single avalanche events do not erase the underlying driving forces or physics (Bak and Chen, 199 1; Noever, 1993). These avalanches can appear when either the effects of single, small-scale events accumulate or when many interactions sum in a complex fashion. For a self-organized system, therefore, the distribution of avalanches spans all sizes and both small and large effects will follow a power-law scaling (Bak et al., 1988a): N(s) = Nos- where values for the exponent 2 and its region of validity are determined experimentally or through numerical simulation. To investigate such complex behaviour in simple systems, the standard sand-pile experiment involves adding single grains of sand to a pre-built pile and recording the weight fluctuations as avalanches carry materials off the balance dish. Various size effects, boundary conditions, grain shape and delivery methods have received critical discussion (Bak and Chen, ; Nagel, 1992). The principal findings centre on the influence of finite size effects of actual experiments (Kadanoff et al., 1989; Held et al., 1990; Evesque, 1991) when compared with periodic or infinite boundary conditions used in numerical simulations (Bak et al. 1988a). Using Biosphere 2 records for atmospheric carbon dioxide, an attractive scientific goal therefore is to test self-organized criticality as a theory governing closed ecosystems. The experimental set-up monitors the carbon dioxide levels as they accumulate around a single probe-site, then at some critical concentration, the system adjusts to maintain short-term balance (Roenneberg et al. 1989). The result is a varying time series with a biological driving force for fluctuations. The metabolism of the closed ecosystem drives C02 changes and external damping (temperature, sunlight, moisture, etc.) determines fluctuation rates. Figure 1 shows an experimental schematic illustrating the basic time series. Therefore to compare the Biosphere 2 directly with previous sand-pile experiments, the environmental regulators (power, growth, sunlight) correspond to the sand grains. Each input can change the C02 reading. The total C02 is equivalent to the weight (internal atmosphere serves as the balance scale). The avalanche or rapid fluctuation in C02 occurs when many inputs sum to give a characteristic atmospheric change. This C02 change corresponds to a sandpile reorganization. This coincides with the critical angle of repose, because a given atmosphere feeds back into a complex Biosphere 2, which sums to produce ecological balance. When monitored across many thousands of events, the statistics of the many-body design follow a power-law scaling across at least three orders of magnitude in event size. The experimental outcome is shown in Figure 2 along with unprocessed atmospheric readings shown in Figure 1. Power-law scaling was found for the 2-year lifecycle of the closed ecosystem. Trace gas analysis determined a total system leak rate of less than 10 per cent. Different atmospheric mechanisms, either as biological or external (anthropogenic), can be observed across the time series, but do not substantially change the slope of the power-law scaling (Figure 3). When the whole data set was reduced across three orders of magnitude in atmospheric levels, the same correlation governs across multiple events. For example, a large oxygen deficit (owing to oxidation of excess soil organics and subsequent concrete absorption of

3 SELF-ORGANISED CRITICALITY IN CLOSED ECOSYSTEMS 599 Time Series For Biosphere Carbon Dioxide Levels Arb. Scale Time Arb. Scale Figure 1. Unprocessed carbon dioxide readings across a 24-month observation period the formed C02) seems not to statistically affect the behaviour of the fluctuation spectrum. The Biosphere 2 seems to provide a robust scaling behaviour driving small- and large-scale fluctuations (Figure 3). The present results do not include how external changes (temperature, sunshine) may influence the statistical power law. Instead what is aimed for is the universal or generic behaviour of a many-body, biological system. The corresponding scaling relation between atmospheric catastrophes and frequency of occurrence is extracted from the carbon dioxide time series. Fluctuations were found to depend on the carbon dioxide concentration for their absolute magnitude, but the slope of the power law can be seen (in Figure 2) to follow a similar scaling relation. Self-similarity in this context implies a scale-free physics where observable phenomenon looks similar at any scale. The signature of such a result is the power-law correlation between small and large ends of the frequency spectrum. In other contexts, spatial self-similarity is described by fractals which capture the fractional dimension of many rivers, mountain ranges, coastlines, soot particles, galaxies, etc. The present self-similarity is temporal because signal fluctuations from the atmospheric time series vary with a critical exponent of f0.05 for at least three orders of magnitude (Figure 3). This value compares with a classical - 1 exponent found for flicker noise (l/fspectrum) to -2.5 found by Held et al. (1 990) in sand-pile experiments. For comparison with open ecosystems, the Byrd climatic record of global C02 over the last 50 ka has a similar power-law relation but with -2.3 as the scaling exponent (Figure 4). With the spectrum determined, the next question arises as to its cause. Figure 2 compares directly the performance of various statistical distributions as they fit the fluctuation spectrum of Biosphere 2 s atmosphere. Log-log values are shown for expectations (perfect fit) based on uniform, Weibull, exponential, and gamma distributions and compared with atmospheric data. A linear relation implies that the underlying cause of the fluctuations shares a basic quality with the assumptions of the governing statistical law. For example, the well-known Gaussian model suggests that fluctuations are random with a bell-shaped signature distributed around the mean. An exponential distribution, on the other hand, is equivalent to Brownian noise, with a kind of thermal randomness which distinguishes (flattens) its fluctuations, particularly at the small-scale end of the spectrum. The gamma distribution is attractive for modellers of chaos generally because it shares some interesting mathematical similarities with simple statistics extracted from unpredictable differential equations. A Weibull distribution is commonly found in systems engineering and can be thought to characterize an ecosystem subject to dependent failures. If one part of the system changes, other parts will rapidly follow suit according to a Weibull mechanism. For the Biosphere 2 data, the best fit is found for either Weibull or uniform (log) distributions. This suggests that many independent random variables are combined in a multiplicative fashion. It is noteworthy to consider the failure of an exponential distribution, because this effectively excludes a thermal (Brownian) cause for atmospheric fluctuations.

4 600 R. J. CRONISE, D. A. NOEVER AND A. BRITTAIN Statistical Distribution for Carbon Dioxide Uniform , Weibull] 1; OO 1 A : Exponential. I Figure 2. Results for proposed statistical distributions governing C02 fluctuations reduced on log-log (base 10) scaling for three orders of magnitude in fluctuations (avalanche events). See text for explanation of physical meaning y= x R = Carbon Dioxide Fluctuations (log) Figure 3. Power-law scaling for fluctuations in carbon dioxide levels. Up to a 100 per cent deviations from the mean does not substantially change the slope of the power-law scaling. Scaling exponent shown, compared with - 1.O for flicker noise and -2.5 for sand-pile models. Inset shows the 95 per cent confidence interval. Most notable is the three orders of magnitude for fluctuations, compared with only a tenfold variation for most typical SOC experiments

5 SELF-ORGAMSED CRITICALITY IN CLOSED ECOSYSTEMS 60 1 (a.u.1 7 I mi A 2 1 y = ~ RA2 = Log-Characteristic C02 Fluctuation Size (a.u.1 Figure 4. Scaling law for Byrd atmospheric carbon dioxide for SO ka. Slope equals -2.3 and suggests Brownian (correlated) noise To compare the finite size effects, the influences of Biosphere 2 size and shape variation, along with the standard deviation for different climates within the Biosphere 2, were not recorded directly for these experiments. For these statistical results, however, the particulars of biogeography appear of secondary influence, because different inputs with markedly different catalytic effects (sunshine, external control of temperature, etc.) do not differ in their avalanche dynamics. As an example, oxygen perturbation in the beginning of the second year seems to have been absorbed within the frequency spectrum without fimdamentally disturbing the Biosphere 2 s statistical signature. The problem of extending the range of fluctuations centres on undercounting small avalanches (owing to instrument sensitivity) and has been observed in sand-pile experiments previously (Kadanoff et al., 1989; Held et al., 1990; Evesque, 1991; Noever, 1991a). Whereas for the Biosphere 2, avalanches were recorded across three orders of magnitude in size (a comparatively large span), the power-law behaviour similarly began to flatten (or undercount) the expectations of the smaller fluctuations relative to the large avalanches (Figure 3). Despite these markedly different inputs, their carbon dioxide spectra shows correlated power laws. The confirmation of these statistical laws in smaller scale test units is a subject for future investigation. To summarize, the present experiments have evaluated a biological analogue to many-body systems previously studied in sand-piles, correlated its long time behaviour to observable statistics and provided a comparative framework for looking at their respective strengths and shortcomings: finite size effects, mechanical versus unautomated delivery system of avalanche particles, and statistical sample size. The difference between the closed ecosystem power law exponent (- 1.3) and the global atmospheric record (-2.30) suggests that the closed system generates flicker noise (llf spectrum) whereas the open system generates Brownian noise ( llf2 spectrum). Physical systems based on flicker noise are moderately correlated, not just over short times but throughout long lifetimes: the system has an apparent statistical memory of prior events linking vastly different scales. Technically, this means that the spectral power contained in large oscillations is the same as the power contained in small ones (differential frequency is linear, dfxf). In other natural contexts, such

6 602 R. J. CRONISE, D. A. NOEVER AND A. BRITTAIN flicker noise has been found in sunspot variations, Earth axial wobbling, undersea air currents, the fluctuating levels for rivers, and so on. Brownian noise, on the other hand is the opposite extreme and the global carbon dioxide spectrum consists of strongly correlated fluctuations. For environmental regulation, it is interesting to notice that the appearance of such scale-free physics is profound. Most physical systems can first be analysed by deciding on which length scales govern its behaviour and ignoring inconsequential or averagable influences. An example is that a theory of hurricane formation need not include a discussion on air molecule collisions except in an average sense. In a self-organized or power-law correlated system, however, such an analysis is not possible because all scales are important. Further, the need for a tunable parameter such as temperature is largely unnecessary because the system self-organizes into a critical state through the natural accumulation of many-sized events. Concrete and physical insight into the meaning that underlies power-law systems lags behind the actual number of its natural occurrence. However, the identification of atmospheric time series as a robust SOC example deserves further attention, particularly when contemplating anthropogenic and natural methods for environmental control. To summarize, the present experiments have evaluated a biological analogue to many-body systems previously studied in sand-piles, correlated its long time behaviour to observable statistics, and provided a comparative framework for looking at their respective strengths and shortcomings. Principal among the experimental difficulties are finite size effects, mechanical versus unautomated delivery of avalanche initiation, and small statistical sample size. Three features distinguish this analysis from previous SOC models: (i) the first large-scale fluctuation data collected from a closed ecosystem; (ii) the large data size (2 years of fluctuations varying over three orders of magnitude); and (iii) the explicit comparison of a fitted statistical distribution (random, exponential, or multiplicative modelling). The second feature, in particular, distinguishes the Biosphere 2 data. The previous best SOC experimental results extended typically over only one order of magnitude before finite size effects began to limit the power law validity. The robustness of the current biological model suggests that many-body dynamics may share some common features even among radically different designs in nature. REFERENCES Babcock, K. L. and Westervelt, R. M Topological melting of cellular domain lattices in magnetic garnet films, fhys. Rev. Lett., 63, Bak, P. and Chen, K Self-organized criticality, Sci. Am., 264, Bak, P. and Tang, C Earthquakes as self-organized critical phenomenon, 1 Geophys. Res., 94, Bak, P., Tang C. and Wiesenfeld, K. 1988a. Self-organized critical phenomenon, fhys. Rev. Lett. 59, Bak, P., Tang, C. and Wiesenfeld, K. 3988b. Self-organized critical phenomenon, fhys. Rev. A, 38, Bak, P. Chen, K. and Tang, C Critical exponents and scaling in self-organized criticality, fhys. Len. A, 147, Brittain, A. M The metabolic diversity, biological activity and stability of the steady state condition in closed ecosystems, PhD dissertation, University of Hawaii, Honolulu, Hawaii. Diodati, P.,Marchesoni F. and Piazza, S Acoustic emission from volcanic rocks: an example of self-organized criticality, Phys. Rev. Lett., 67, Evesque, I? Analysis of the statistics of sandpile avalanches using soil mechanics: results and concepts, fhys. Rev. A, 43, Held, G. A., Solina, D. H., Keane, D. T., Haag, W. J., Horn, I? M. and Grinstein, G Experimental study of critical mass fluctuations in an evolving sandpile, Phys. Rev. Lett. 65, Kadanoff, L. I?, Nagel, S. R., Wu, L. and Su-Min Zhou Scaling and universality in sandpiles, fhys. Rev. A, 39, Keams, E. A Energetics of materially closed ecosystems, PhD dissertation, University of Hawaii, Honolulu, Hawaii. Nagel, S Instabilities in a sandpile, Rev. Mod. fhys., 64, Noever, D. A Himalayan sandpiles, fhys. Rev. A, 47, Noever, D. A. 1991a. Bioconvective patterns, synchrony and survival, fhys. Rev Lett., 65, Noever, D. A. 1991b. Evolution of bioconvective patterns in variable gravity, fhys. Rev A, 44, Obenhuber, D. C The persistence of life as measured by carbon cycling in closed ecological systems, PhD dissertation, University of Hawaii, Honolulu, Hawaii. Roenneberg, T., Colfax, G. N. and Hastings, J Circadian rhythms in Gonyaulwr, J: Biol. Rhythms, 4, Schwa, K. W Evidence for organized small scale structures in fully developed turbulence, fhys. Rev. Lett., 64, Shaffer, J. A Stability in closed ecological systems: an examination of material and energetic pammeters, PhD dissertation, University of Hawaii, Honolulu, Hawaii.