Simple spatially distributed model of the global carbon cycle and its dynamic properties

Abstract

A specially simplified spatially distributed model of the global carbon cycle (GCC) is presented that is allowed to get a general solution and to study its basic dynamic properties (like spectral portrait, composition and decomposition of solutions, etc.). Assuming that the productivity depends linearly on concentration of carbon in the atmosphere, the problem is reduced to a standard linear one. In order to exclude the oceanic part of the GCC a parameterisation based on a principal difference of characteristic times between the terrestrial and oceanic parts is used. For calibration the Bazilevich data containing information on plant productivity, living biomass and soil organic matter of different biomes is used. Calculating the matrix eigenvalues of the basic linear problem we get its spectral portrait. It shows that there are neither specific characteristic times nor a fibering of phase space, which could allow to construct correct asymptotics: characteristic times change sufficiently smoothly from several years to several hundreds. Among all the exponentially damped solutions there are three oscillating ones with very long periods: from 117 to 4000 years. This portrait is very stable in relation to variations in the areas, covered by different biomes. Therefore, the terrestrial part of the GCC is a well-structured dynamic system with very wide opportunities to compensate disturbances with different time characteristics. Note that the general solution is very sensitive to the spatial distributions of productivity, living biomass and soil organic matter but all these patterns are uncertain. Certainly, the atmosphere is a natural integrator of local carbon flows, and we can hope that then different uncertainties partially compensate one another, when we estimate the total amount of carbon in the atmosphere. On the other hand, any distribution is described by its statistical moments and namely these moments are very robust in respect to numerical meanings of the distributed value. A special operation of convolution to the general solution of a spatially distributed model is suggested. This allows getting a dependence of convoluted solution on first moments such as means and covariances. However, it is necessary to note that instead of the distributions of living biomass and dead organic matter in soils the distributions of inverse residence times of carbon in different types of vegetation and soils have to be considered. One can see that non-homogeneity of these distributions on the whole increases the amount of carbon in the atmosphere. If comparing our prognosis with the prognoses of others with using more ‘complex’ and ‘geographically explicit’ models, then the difference will be incidental.