Model description

The model is based on the Caldeira/Kasting model [1], which is extended by a geodynamic model of the geosphere. It consists of the following compartments: geosphere, atmosphere, and biosphere, which are linked through nonlinear feedbacks. An external forcing is given by the increasing solar luminosity.

• Climate. The climate of the planet is described by the energy balance equation for the surface temperature:

   

  where is the greenhouse warming which is a monotonous increasing function of the CO content in the atmosphere , i.e. .

  Insolation S(t) increases in time:

   

  For constant we get from Eq. 1 temperatures below C in the past in contradiction to geologic records (Faint Young Sun paradoxon). A solution of this paradox can be found if the intrinsic feedback mechanism of the carbonate-silicate cycle is taken into account.

• Carbon cycle.

    
  Figure 1: Global carbon cycle
  

  Balance equation between sources and sinks of the carbon in the atmosphere for geologic times (see Fig. 1) yields

   

  where is the weathering rate (per area) normalized to the present day value (sink of carbon), the normalized continental area and the normalized spreading rate (source of carbon). The equation can be written as

   

  where GFR is the geophysical forcing ratio. The total amount of carbon in the Earth system can be roughly approximated to 107 ppm.

• Weathering rate. Carbonate-silicate rock weathering depends on the temperature and the carbon in the soil:

   

  where is the activity of in the soil. is a function in the following linear form:

   

  denotes the biologic productivity. Therefore, the biosphere increases the weathering rate of carbon.

• Biologic productivity. is a function of the surface temperature and atmospheric CO :

   

  For we have a parabolic form, for a Michaelis-Menten hyperbola (see Fig. 2).

    
  Figure 2: Biologic productivity as a function of T and .
  

• Geodynamics. A basic assumption of the Caldeira-Kasting model is a static geosphere, i.e.

   

  This so-called geostatic model (GSM) is a rather rough approximation. A geodynamic model (GDM) with spreading and continental growth give us a significant improvement.

• Continental growth. For continental growth a data-based model (see Fig. 3) was used [2].

    
  Figure 3: Normalized value of the continental area.
  

• Spreading rate. According to the boundary layer theory the spreading rate is given as a function of the heat flow calculated by the cooling process of an oceanic plate [3]:

   

  The mantle heat flow and temperature are determined by a parameterized convection model.

  Using Eq. 4 as the initial equation and using Eqs. 1,2 and 5-7 the coevolution of the geosphere-biosphere system can be described by solving

   

  The corresponding temperature is calculated using Eq. 1.