These models solve coupled, prognostic, non-linear, partial differential equations on a three-dimensional grid for both the atmosphere and the ocean. The coupled equations are the consequence of the following physical laws: Newton's second law, mass conservation, first and second law of thermodynamics, equations of state for air and ocean water, budgets of air humidity, liquid water in the atmosphere, in soils and rivers as well as cloud ice and snow, radiation laws of Planck and Kirchhoff, radiative transfer equation, spectral absorption by gases involving some quantum mechanics (see also box 3.1). In addition, empirical relations between vegetation, bare surfaces and radiation parameters are needed. Another complication in solving these equations arises from the changed surface characteristics during the time integration, e.g. due to snowfall or snowmelt. Many physical processes have to be parameterized, because the spatial resolution of the models cannot resolve the manifold sub-grid scale processes like the influence of small clouds or even cloud ensembles. Such parameterizations are often derived from dedicated field experiments.

Box 3.1: Equations and Empirical Relations in a Climate Model

All the equations shown are simplified ones as the full equations would, for example, also contain sound waves, whose high speed would force the numerical scheme to extremely small time steps for the solution of the prognostic equations.

All these equations relate the key variables like velocity vector V , pressure p, acceleration of gravity g, rotational speed of the Earth i2 , temperature T, density p and absolute humidity with diabatic (i.e. entropy generating) processes (in rectangles) originating from friction at the surface Fvj>, net radiation flux divergence Q, surface heat flux FT, phase fluxes of water Sq and humidity fluxes at the surface.

Ocean MPI-OM

Figure 3.1 Schematic diagram of the climate model components used for typical climate change runs. The general circulation model of the atmosphere also includes land surface processes like evapotranspiration, river run-off and the ocean module contains sea ice thermodynamics and rheology. Volcanism as well as solar radiation changes is handled as external processes like atmospheric trace substance concentration changes due to anthropogenic activities (MPI, 2006b)

(a) CARTESIAN GRID GCM

(a) CARTESIAN GRID GCM

Figure 3.2 Schematic representation of the 3-dimensional grid of a climate model

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