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Next: Acknowledgements Up: Reduction of Biosphere Life Previous: Weathering and continental growth

Results and discussion

In the extensive simulations we have conducted, the parameters for the present state of the Earth system (e.g. tex2html_wrap_inline1498) are used as starting values. By combining Eqs. 2 to 6 with Eq. 1, a relation is found which depends only on the surface temperature tex2html_wrap_inline1300, and the atmospheric COtex2html_wrap_inline1266 content tex2html_wrap_inline1322. The next step is to use the greenhouse model of Caldeira and Kasting[1] in order to express tex2html_wrap_inline1300 in terms of tex2html_wrap_inline1322. The resulting equation contains only tex2html_wrap_inline1322 as an unknown variable. The root of this equation yields the equilibrium solution of the atmospheric COtex2html_wrap_inline1266 content, tex2html_wrap_inline1322. Finally, the equilibrium values of tex2html_wrap_inline1300, tex2html_wrap_inline1292, and tex2html_wrap_inline1318 can be calculated. This is the solution for the present state of our Earth system.

To perform the calculations at any other time, one has to use the time-dependent solar insolation (Eq. 3) and a possible change of the dimensionless weathering rate. The value of tex2html_wrap_inline1522 can be determined from the ratio of the dimensionless mid-ocean sea-floor spreading rate and the dimensionless continental area from Franck and Bounama[30, 31] via Eq. 10. Applying the procedure described above, we run our model back to the Proterozoic. At this geological era, life had already changed from anaerobic to aerobic forms, so from that time on the biological productivity can be described by Eq. 6. Starting from the present state again, we run our model 1.5 Ga into the future. This is done by extrapolating the present day continental-growth rate and by calculating the spreading rate according to Franck and Bounama[31]. In addition, we performed the whole procedure also for different orbital distances between Earth and Sun.

The results for the mean global surface temperature tex2html_wrap_inline1300 and the normalized biological productivity tex2html_wrap_inline1528 are plotted in Figs. 2 and 3, respectively. Fig. 2 compares the evolution of the mean global surface temperature from the Proterozoic up to the planetary future for the geostatic model (GSM) and for our geodynamical model (GDM). Up to 1 Ga into the future, the temperature varies only within an interval of tex2html_wrap_inline1532 to tex2html_wrap_inline1534. This stabilization of the surface temperature is a result of the carbonate-silicate self-regulation within the Earth system with respect to growing insolation as an external forcing.

  figure255
Figure 2: Past and future variation of the mean global surface temperature tex2html_wrap_inline1536 for the four employed models: GSM-asymptotic (+), GDM-asymptotic (tex2html_wrap_inline1540), GSM-parabolic (tex2html_wrap_inline1542), and GDM-parabolic (tex2html_wrap_inline1544). The two parabolic models have multiple solutions in the past. The horizontal dashed line indicates the ``optimal'' biogeophysical temperature, i.e., tex2html_wrap_inline1546.

In contrast to the GSM approach, the GDM scheme shows a temperature stabilization at a higher level in the past and a lower one in the future (see Fig. 2). This result is found for the asymptotic biospheric productivity (Eq. 7) as well as for the parabolic one described by Eq. 8. Values before 2 Ga ago are not shown because at these times the GDM gives atmospheric COtex2html_wrap_inline1266 concentrations higher than tex2html_wrap_inline1360 ppm. Under the latter conditions the greenhouse model employed is not valid. Nevertheless, both models show a good stabilization of the mean global surface temperature tex2html_wrap_inline1300 against increasing insolation. In the far future near 1 Ga from now, all curves converge and show a strong increase in tex2html_wrap_inline1300 to almost tex2html_wrap_inline1432. All higher forms of life, especially the photosynthesis-based biosphere, will certainly be extinguished at this time. This kind of sensitivity was also discussed for other continental-growth models in previous papers [31, 23].

During the long-term evolution of the Earth system, the development of the normalized biological productivity (Fig. 3) shows a remarkably different behaviour for the asymptotic and the parabolic models, respectively.

  figure268
Figure 3: Past and future variation of the normalized biological productivity tex2html_wrap_inline1528 for our four Earth system models. Note that the two parabolic models have multiple solutions in the past (see also Fig. 2). The dashed branch lines correspond to backward-directed evolutions.

Because of the chosen set of parameters used in Eq. 8, both parabolic models (GSM and GDM) have zero productivities for times earlier than 500 Ma before present. This obviously contradicts the geological record. Therefore, in the following, we will restrict ourselves to asymptotic models only. In the past, these models generate higher values of tex2html_wrap_inline1528 than today. Our favoured model, the GDM-asymptotic model, calculates twice the present biological productivity for most of the Proterozoic. This seems reasonable because of the higher mean global surface temperatures at that time. As continental growth starts late in our GDM-asymptotic model, the latter generates a strong increase in the biological productivity in the Proterozoic. This model also provides the shortest life span of the photosynthesis-based biosphere - nearly 300 Ma shorter than the corresponding GSM-asymptotic model!

  figure274
Figure 4: Evolution of the atmospheric carbon content as described by the models GSM-asymptotic (red line) and GDM-asymptotic (green line). The terrestrial life corridor (Eq. 14) is identical to the non-dashed region. The plotted isolines are the solutions of GSM-asymptotic for the indicated fixed values of the normalized weathering rate tex2html_wrap_inline1522.

Fig. 4 shows the atmospheric carbon content over geologic time from the Hadean to the planetary future for the two asymptotic models. In the dashed region of Fig. 4 no photosynthesis is possible because of inappropriate temperature or atmospheric carbon content. The non-dashed region is the ``terrestrial life corridor (TLC)''. Formally, the life corridor in the (tex2html_wrap_inline1322, tex2html_wrap_inline1300)-domain is defined as
 equation283
For the asymptotic model we have (inserting Eqs. 6, 7, and 9 into Eq. 13):
 equation292
It is possible to map TLC in different parameter spaces. For instance, tex2html_wrap_inline1300 can be substituted with the help of the greenhouse model by tex2html_wrap_inline1322. Therefore, TLC can be depicted in the (tex2html_wrap_inline1322, t)-domain, as shown in Fig. 4. The reference model GSM is based on a weathering rate that is always equal to the present-day rate tex2html_wrap_inline1584. The GDM takes into account the influence of a growing continental area and the changing spreading rate on weathering. The GDM has higher weathering rates for the past (i.e. tex2html_wrap_inline1586). This can be explained easily with the help of Eq. 10, because in the geological past we had higher spreading rates tex2html_wrap_inline1588 and a smaller continental area tex2html_wrap_inline1590. In the planetary future we find the reversed situation: lower spreading rates and higher continental area will reduce the atmospheric carbon content, and this is the reason why the biosphere`s life span is significantly shorter because photosynthesis can persist only down to the critical level of 10 ppm atmospheric COtex2html_wrap_inline1266 concentration. As mentioned above, the GDM model does not work at atmospheric COtex2html_wrap_inline1266 concentrations higher than tex2html_wrap_inline1360 ppm, and therefore the model fails on time scales of about 2 Ga ago. Compared to the smooth curve of the GSM-asymptotic model, our favoured model (GDM-asymptotic) provides a curve with a certain structure that is directly related to the step-like continental growth shown in Fig. 1.

  figure310
Figure 5: Evolution of the atmospheric carbon content as described by the models GSM-parabolic (red line) and GDM-parabolic (green line). The terrestrial life corridor for the parabolic model is defined by tex2html_wrap_inline1600 and is identical to the non-dashed region. The plotted isolines have the same meaning as in Fig. 4.

The life corridor for the two parabolic models is plotted in Fig. 5. As already mentioned above, these models do not work very well for the past because the upper boundary for the atmospheric COtex2html_wrap_inline1266 content is rather low, between tex2html_wrap_inline1604 and tex2html_wrap_inline1606 ppm. Nevertheless, both curves show the phenomenon of bistability in the Phanerozoic. Dependent on the time direction, there are different evolutionary paths in the diagram. Such a hysteresis, in a much more pronounced form, can also be calculated for the Daisyworld model class [20]. Similar effects were discussed by Kump and Volk[18]. They found both bistability and insufficient conditions for life for times of more than 100 Ma ago for the parabolic model, which is clearly not realistic.

Besides calculating the TLC, i.e., the evolution of atmospheric carbon regimes supporting photosynthetis-based life in time, we calculated the behaviour of our virtual Earth system at various distances from the Sun, using different insolations. Hart[34, 35] calculated the evolution of the terrestrial atmosphere over geologic time by varying the distance from the Sun. In his approach, the habitable zone (HZ) is the region within which an Earth-like planet might enjoy moderate surface temperatures needed for advanced life forms. Kasting et al.[40] defined the HZ of an Earth-like planet as the region where liquid water is present at the surface. According to this definition the inner boundary of the HZ is determined by the loss of water via photolysis and hydrogen escape. Kasting et al.[40] propose three definitions of the outer boundary of the HZ. All of them are connected to a surface-temperature limit of 273 K. This is in agreement with our definition of the low-temperature boundary of the biological productivity.

According to Kasting et al.[40] the outer boundary of the HZ is determined by COtex2html_wrap_inline1266 clouds that attenuate the incident sunlight by Rayleigh scattering. The critical COtex2html_wrap_inline1266 partial pressure for the onset of this effect is about 5 to 6 bar (Kasting, personal communication, 1999). On the other hand, the effects of COtex2html_wrap_inline1266 clouds have been challenged recently by Forget and Pierrehumbert[28]. COtex2html_wrap_inline1266 clouds have the additional effect of reflecting the outgoing thermal radiation back to the surface. In this way, they could have extended the size of the HZ in the past.

Hart[34, 35] found that the HZ between runaway greenhouse and runaway glaciation is surprisingly narrow for G2 stars like our Sun: tex2html_wrap_inline1622 AU, tex2html_wrap_inline1624 AU (AU = astronomical unit). A main disadvantage of these calculations is the neglect of the negative feedback between atmospheric COtex2html_wrap_inline1266 content and mean global surface temperature discovered later by Walker et al.[7]. The implementation of this feedback by Kasting et al.[39] provides an almost constant inner boundary (tex2html_wrap_inline1628AU) but a remarkable extension of the outer boundary beyond Martian distance (tex2html_wrap_inline1630 AU). Later Kasting et al.[40] and Kasting et al.[38] recalculated the HZ boundaries as tex2html_wrap_inline1628 AU and tex2html_wrap_inline1634 AU.

In a recent paper, Williams and Kasting[53] investigated the problem of habitable planets with high obliquities with the help of an energy-balance climate model. In their two-dimensional model tex2html_wrap_inline1636 shifts to tex2html_wrap_inline1638 AU. In our approach, the HZ for an Earth-like planet is the region within which the biological productivity is greater than zero. This is the region where the surface temperature stays between tex2html_wrap_inline1640 and tex2html_wrap_inline1642 and the atmospheric COtex2html_wrap_inline1266 content is higher than 10 ppm suitable for photosynthetis-based life. The term ``Earth-like'' explicitly implies also the occurrence of plate tectonics. In order to assess the HZ, the life corridor is calculated by varying the distance of our Earth-like planet to the Sun for various time steps in the past, the present, and the future. Fig. 6 shows the results for the present state of the Earth system.

  figure345
Figure 6: Present habitable zone (non-dashed region) for GSM-asymptotic (red) and GDM-asymptotic (green). The two models provide identical answers as they are both fitted to the same present-day parameters. Present inner and outer boundaries are determined as tex2html_wrap_inline1648 AU and tex2html_wrap_inline1650 AU, respectively.

In this case, GSM and GDM give identical curves because both models are fitted to the same present parameters. We find tex2html_wrap_inline1648 AU, so our Earth system is even closer to a runaway greenhouse effect than in the calculations of Hart[35] and Kasting et al.[39]. The outer boundary is extended in comparison to the model of Hart[35], but not beyond the Martian distance (tex2html_wrap_inline1650AU). Despite the different ansatz, our results are comparable to those found by Kasting et al.[40], Kasting et al.[38], and Williams and Kasting[53].

  figure359
Figure 7: The habitable zone (non-dashed region) for an Earth-like planet in the ``near'' future (0.5 Ga). For GDM-asymptotic (green line) the respective boundaries are tex2html_wrap_inline1656 AU and tex2html_wrap_inline1658 AU. Because tex2html_wrap_inline1660 will exactly coincide with the actual distance of planet Earth from the Sun (1 AU), the life span of the biosphere will end at this time. For GSM-asymptotic (red line) the respective boundaries are tex2html_wrap_inline1662 AU and tex2html_wrap_inline1664 AU.

In the planetary future at 0.5 Ga from now, the situation has changed dramatically (see Fig. 7: the geodynamic Earth-system model GDM predicts that the lower boundary for atmospheric COtex2html_wrap_inline1266 concentration has been reached already (tex2html_wrap_inline1670 AU), and the boundary for a runaway glaciation has moved significantly inward (tex2html_wrap_inline1658 AU). Our results for the estimation of the HZ for all times are summarized in Fig. 8, where we have plotted the width and position of the HZ for the GSM and GDM variants over time.

  figure372
Figure 8: Evolution of the habitable zone (HZ) for GSM-asymptotic (red line) and GDM-asymptotic (green line). Note that for GSM-asymptotic the HZ has a slightly increasing width and shifts outward from the Sun. The HZ for our favoured model, GDM-asymptotic, is both shifting and narrowing over geologic time, terminating life definitely at 1.4 Ga. The optimum position for an Earth-like planet would be at tex2html_wrap_inline1674 AU. In this case the life span of the biosphere would realize the maximum life span, i.e. the above-mentioned 1.4 Ga. This figure is the main finding of our investigation.

For the geostatic case (GSM) the width of the HZ slightly increases and shifts outward over time. In about 800 Ma the inner boundary tex2html_wrap_inline1660 reaches the Earth distance from the Sun (R = 1 AU) and the biosphere ceases to exist, as already found above for this model. Our geodynamic model shows both a shift and a narrowing of the HZ: the inner boundary tex2html_wrap_inline1684 reaches the Earth distance in about 500 Ma from now in correspondence with the shortening of the life span of the biosphere by about 300 Ma as compared to the geostatic model. In the GDM, the outer boundary tex2html_wrap_inline1636 decreases in a strong nonlinear way. This result is in contrast to the GSM and to the results of Kasting et al.[40] and Kasting[38]. Since our criteria for the HZ is defined using biological productivity alone, the critical boundaries can be extended for the temperature from tex2html_wrap_inline1642 to tex2html_wrap_inline1432 or higher. But our results show that the inner boundary of the HZ is determined by the 10 ppm limit (Fig. 6 and 7) and the outer boundary by the tex2html_wrap_inline1640 limit.

Of course, there may exist chemolithoautotrophic hyperthermophiles that might survive even in a future of higher temperatures, rather independently of atmospheric COtex2html_wrap_inline1266 pressures. But all higher forms of life would certainly be eliminated under such conditions. Our biosphere model is actually only relevant to photosynthesis-based life. Therefore, in the time span under consideration, the upper temperature does not affect the results for TLC and HZ, respectively.

The main objective of our paper is to generate model results for the dependence of the biosphere life span on the sea-floor spreading rate and the history of continent growth. Therefore, sensitivity tests based on the variations of these geodynamics entities have been performed. In Fig. 9, the ensemble of alternative continental-growth models is sketched. The linear and the delayed growth models reflect the simplest theoretical assumptions [31]. The model of Reymer and Schubert[46] is based on the assumption of an approximately constant continental freeboard. For all of these models surface temperature, COtex2html_wrap_inline1266 history, and biological productivity have been already investigated by Franck et al.[23] in a recent study. Furthermore, we added a model with constant continental area which seems to be in contrast to geological records. Therefore, this geodynamic scenario is the most unrealistic one. In all cases, the spreading rates are calculated with the help of Eq. 11.

  figure392
Figure 9: Normalized continental area tex2html_wrap_inline1702 as a consequence of the following continental-growth models: (a) delayed growth, (b) linear growth, (c) Reymer and Schubert, (d) constant area. The thick line indicates the Condie model (see Fig. 1).

Fig. 10 summarizes the results for the HZs for the additional four continental-growth scenarios indicated in Fig. 9. It is obvious that in all cases the life span of the biosphere is significantly shorter than in the GSM and the magnitude remains approximately the same for all alternatives (Table 1). The optimum distances for an Earth-like planet vary slightly between 1.09 and 1.21 AU. The maximum life span at the optimum distance varies between 1.35 and 2.05 Ga in cases a, b, and c, respectively, evidently increasing with decreasing continental-growth rate. In the most unrealistic case without continental growth but geodynamics, i.e. scenario d, the maximum life span is extended to 2.98 Ga.

  figure400
Figure 10: Habitable zones of the GDM-asymptotic for the four alternative continental-growth models: (a) delayed growth, (b) linear growth, (c) Reymer and Schubert, (d) constant area. Obviously, the so-calculated HZs do not differ in their qualitative behaviour. The biosphere life span varies only slightly in magnitude and is always shorter than in the geostatic case (GSM). Exact values for the biosphere life span, the optimum distance, and the maximum life span, respectively, can be found in Table 1.

 
Continental growth model Biosphere life span (Ga) tex2html_wrap_inline1714 (AU) Maximum life span (Ga)
Delayed growth 0.48 1.09 1.35
Condie 0.48 1.08 1.40
Linear growth 0.53 1.11 1.80
Reymer and Schubert 0.58 1.13 2.05
Constant area 0.63 1.19 2.98
Table 1: Life span of the biosphere, optimum distance to the Sun, tex2html_wrap_inline1714, and maximum life span at tex2html_wrap_inline1714, respectively, for the five different continental-growth models.

 

The comparison of Figs. 8 and 10 demonstrates that the character of of the HZ remains similar in all the GDM cases. There are only minor differences in the quantitative results (see Table 1). The results for the geodynamic model with continental growth according to Condie[27] are well within the range of the overall model ensemble. Therefore, we favour the Condie continental-growth models as it is directly related to the geological record.

Our results for life corridors and habitable zone are certainly relevant for the overall debate on sustainable development and geocybernetics (see, e.g., [47]). Regarding the ``ecological niche for civilization'' on Earth, we can learn from Fig. 8 that within the framework of our favoured model (GDM-asymptotic) the ``optimal'' planetary distance from the Sun would be about 1.08 AU. At such a distance the self-regulation mechanism would work ideally against external forcing arising from increasing solar insolation or other perturbations, and the life span of the biosphere would be extended to 1.4 Ga. But after that time, the biosphere will definitely cease to exist. Our findings are also relevant to the search for habitable zones around other main-sequence stars [35].

Finally, we want to emphasize that all predictions about the long-scale evolution of the Earth system include uncertainties. These uncertainties are mainly related to inherent imperfections in modelling single components of the Earth system. Nevertheless, we are convinced that our main conclusions, especially about the shortening of the life span and the narrowing of the HZ, are qualitatively correct. A more detailed analysis of the response of the Earth system against perturbations at various time scales requires a dynamic extension of the present models, however. Such an investigation needs particularly the analysis of possible accelerations and inertial effects of the global weathering rate. This will be part of future work.


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Next: Acknowledgements Up: Reduction of Biosphere Life Previous: Weathering and continental growth

Werner von Bloh (Data & Computation)
Mon Jul 10 15:15:10 MEST 2000