## Initial conditions boundary conditions and model parameters

Mathematically speaking, a climate model is a system of equations with diagnostic and prognostic equations. Diagnostic equations instantaneously link different variables to each other, for example, the hydrostatic assumption linking pressure and density. Prognostic equations need to be integrated forward in time, for example, the Navier-Stokes equations describing hydrodynamics. In theory, it should be possible to integrate prognostic equations backwards in time, but this task is almost insurmountable because many diagnostic relationships are nonbijective (i.e. cannot easily be inverted). The model is thus integrated forward in time and one must define initial conditions, boundary conditions, and parameters.

• Initial conditions represent the climate state from which the model equations are integrated. They are generally supplied from a previous experiment or from observations.

• Boundary conditions define values or fluxes at the boundary of the domain, such as the surface topography and incoming shortwave radiation at the top of the atmosphere. The latter is calculated from the geometry of the Earth's orbit.

• Parameters are constants used in the equations. Some are directly specified from laboratory experiments (e.g. heat capacity of water) or direct observations (albedo of sea-ice, concentrations of greenhouse gases). Others are more phenomenological and they must be calibrated by comparing model results with observations (see below).

Once initial conditions, boundary conditions, and parameters are specified, the model is run to perform either an steady-state experiment or a transient experiment.

The climate modeler's "steady-state experiment" is an integration for which boundary conditions and parameters are constant, except for the insolation seasonal cycle. Such experiments are suited to study the statistics of a quasi-steady climate, where the time-scale of the external forcing change is larger than the longest dissipative time-scale of the system. This is about a few thousand years if deep-ocean dynamics are taken into account (this number is reached by dividing the depth of the ocean to the square by the turbulent vertical diffusion coefficient). Initial conditions seem unimportant in steady-state experiments because they are eventually dissipated ("forgotten"). There are two caveats to this statement. First, it is more economical to guess initial conditions that are not too far from the solution in order to reduce as much as possible the time spent by the system to reach it (spin-up time). Second, there may be two quasi-state solutions to the model equations, with only a small probability of transition from the one to the next: For example, a green and a white Sahara (Claussen, this volume). An experiment is said to be transient if the boundary conditions and some other parameters (such as greenhouse gas concentrations) change during the integration. These changes constitute an external forcing that influences the climate trajectory. Two categories of transient experiments may be distinguished. In the first one, the external forcing varies slowly or at a rate comparable to the dissipative time-scales of the system. A good example is the change in the spatial and seasonal incoming solar radiation due to the quasi-cyclic variations in the Earth's orbit: eccentricity, precession, and tilt of the rotation axis on the ecliptic (i.e. orbital forcing). Such experiments allow us to identify and understand nonlinear processes in the response to the forcing, such as deglaciation or desertification of the Sahara (Claussen et al. 1999). The response is said to be abrupt if its characteristic time-scale is shorter than that of the forcing (Alley et al. 2003). The second category of transient experiments gathers those in which the forcing varies with a characteristic time shorter than the dissipative time-scales of the system. In this case the abrupt character of the response is due to the forcing itself. Experiments of this kind are usually performed to test hypothetical scenarios such as a discharge of freshwater in the ocean (Renssen et al. 2001; Bauer et al. 2004; LeGrande et al. 2006).

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