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We quantify the long-term predictability of global mean daily temperature data by means of the Renyi entropy of second order K-2. We are interested in the yearly amplitude fluctuations of the temperature. Hence, the data are low- pass filtered. The obtained oscillatory signal has a more or less constant frequency, depending on the geographical coordinates, but its amplitude fluctuates irregularly. Our estimate of K-2 quantifies the complexity of these amplitude fluctuations. We compare the results obtained for the CRU data set (interpolated measured temperature in the years 1901- 2003 with 0.5 degrees resolution, Mitchell et al., 2005(1)) with the ones obtained for the temperature data from a coupled ocean-atmosphere global circulation model (AOGCM, calculated at DKRZ). Furthermore, we compare the results obtained by means of K-2 with the linear variance of the temperature data

We propose a new approach to calculate recurrence plots of multivariate time series, based on joint recurrences in phase space. This new method allows to estimate dynamical invariants of the whole system, like the joint Renyi entropy of second order. We use this entropy measure to quantitatively study in detail the phase synchronization of two bidirectionally coupled chaotic systems and identify different types of transitions to chaotic phase synchronization in dependence on the coupling strength and the frequency mismatch. By means of this analysis we find several new phenomena, such a chaos-period-chaos transition to phase synchronization for rather large coupling strengths. (C) 2004 Elsevier B.V. All rights reserved

Extreme weather events are likely to occur more often under climate change and the resulting effects on ecosystems could lead to a further acceleration of climate change. But not all extreme weather events lead to extreme ecosystem response. Here, we focus on hazardous ecosystem behaviour and identify coinciding weather conditions. We use a simple probabilistic risk assessment based on time series of ecosystem behaviour and climate conditions. Given the risk assessment terminology, vulnerability and risk for the previously defined hazard are estimated on the basis of observed hazardous ecosystem behaviour.
We apply this approach to extreme responses of terrestrial ecosystems to drought, defining the hazard as a negative net biome productivity over a 12-month period. We show an application for two selected sites using data for 1981-2010 and then apply the method to the pan-European scale for the same period, based on numerical modelling results (LPJmL for ecosystem behaviour; ERA-Interim data for climate).
Our site-specific results demonstrate the applicability of the proposed method, using the SPEI to describe the climate condition. The site in Spain provides an example of vulnerability to drought because the expected value of the SPEI is 0.4 lower for hazardous than for non-hazardous ecosystem behaviour. In northern Germany, on the contrary, the site is not vulnerable to drought because the SPEI expectation values imply wetter conditions in the hazard case than in the non-hazard case.
At the pan-European scale, ecosystem vulnerability to drought is calculated in the Mediterranean and temperate region, whereas Scandinavian ecosystems are vulnerable under conditions without water shortages. These first model- based applications indicate the conceptual advantages of the proposed method by focusing on the identification of critical weather conditions for which we observe hazardous ecosystem behaviour in the analysed data set. Application of the method to empirical time series and to future climate would be important next steps to test the approach.

The maximum population, also called Earth's carrying capacity, is the maximum number of people that can live on the food and other resources available on planet Earth. Previous investigations estimated the maximum carrying capacity as large as about 1 trillion people under the assumption that photosynthesis is the limiting process. Here we use a present state-of-the-art dynamic global vegetation model with managed planetary land surface, Lund-Potsdam-Jena managed Land (LPJmL), to calculate the yields of the most productive crops on a global 0.5 degrees x 0.5 degrees grid. Using the 2005 crop distribution the model predicts total harvested calories that are sufficient for the nutrition of 11.4 billion people. We define scenarios where humankind uses the whole land area for agriculture, saves the rain forests and the boreal evergreen forests or cultivates only pasture to feed animals. Every scenario is run in an extreme version with no allowance for urban and recreational needs and in two soft versions with a certain area per person for non-agricultural use. We find that there are natural limits of the maximum carrying capacity which are independent of any increase in agricultural productivity, if non-agricultural land use is accounted for. Using all land planet Earth can sustain 282 billion people. The save-forests-scenario yields 150 billion people. The scenario that cultivates only pasture to feed animals yields 96 billion people. Nevertheless, we should always have in mind that all our calculated numbers for the carrying capacity refer to extreme scenarios where humankind may only vegetate on this planet. Our numbers are considerably higher than the general median estimate of upper bounds of human population found in the literature in the order of 10 billion.

We have undertaken a thorough dynamical investigation of five extrasolar planetary systems using extensive numerical experiments. The systems Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208 were examined concerning the question of whether they could host terrestrial-like planets in their habitable zones (HZ). First we investigated the mean motion resonances between fictitious terrestrial planets and the existing gas giants in these five extrasolar systems. Then a fine grid of initial conditions for a potential terrestrial planet within the HZ was chosen for each system, from which the stability of orbits was then assessed by direct integrations over a time interval of 1 million years. For each of the five systems the 2-dimensional grid of initial conditions contained 80 eccentricity points for the Jovian planet and up to 160 semimajor axis points for the fictitious planet. The computations were carried out using a Lie-series integration method with an adaptive step size control. This integration method achieves machine precision accuracy in a highly efficient and robust way, requiring no special adjustments when the orbits have large eccentricities. The stability of orbits was examined with a determination of the Renyi entropy, estimated from recurrence plots, and with a more straightforward method based on the maximum eccentricity achieved by the planet over the 1 million year integration. Additionally, the eccentricity is an indication of the habitability of a terrestrial planet in the HZ; any value of e > 0.2 produces a significant temperature difference on a planet's surface between apoapse and periapse. The results for possible stable orbits for terrestrial planets in habitable zones for the five systems are: for Gl 777 A nearly the entire HZ is stable, for 47 Uma, HD 72659 and HD 4208 terrestrial planets can survive for a sufficiently long time, while for Gl 614 our results exclude terrestrial planets moving in stable orbits within the HZ. Studies such as this one are of primary interest to future space missions dedicated to finding habitable terrestrial planets in other stellar systems. Assessing the likelihood of other habitable planets, and more generally the possibility of other life, is the central question of astrobiology today. Our investigation indicates that, from the dynamical point of view, habitable terrestrial planets seem to be compatible with many of the currently discovered extrasolar systems