Initial conditions and forcing

In earlier chapters we have found that it is necessary to specify conditions on the boundaries of a problem domain in order to obtain a solution for parameters within the domain. Vertical velocities at the surface were specified in Chapter 5, temperatures and temperature gradients in Chapter 6, and stresses and velocities in Chapter 10. In all of these examples, the solutions we sought were steady-state solutions, so all time derivatives were 0.

In many modeling studies, time-dependent solutions are desired. Indeed one of the strengths of numerical modeling is that we can study the evolution of a complex system through time - a goal that is only rarely achievable with analytical solutions (Equation (6.31) is such an exception). In time-dependent models it is necessary to specify conditions at time t = 0, called initial conditions, and also, in most cases, the temporal evolution of some of these conditions, usually those at the surface. The latter is frequently referred to as the forcing. We have already encountered an example of an initial condition earlier in this chapter, in discussing the solution to Equation (11.8).

The choice of initial conditions depends on the extent of our knowledge of those conditions. If there is no well-defined condition from which to start the integration, it can be started with an unrealistic situation such as a temperature profile that varies linearly with depth or an ice sheet with a parabolic profile. This approach is particularly appropriate in problems involving cyclical changes, such as seasonal changes in temperature at the surface or changes in climate driven by variations in Earth's orbit -the Milankovitch cycles. The model would then be run through several cycles until the solution at a given point in a cycle is essentially identical to that at the same point in the previous cycle. One can then conclude that the model has "forgotten" the unrealistic initial conditions. This procedure is commonly referred to as a spin up of the model. The final solution can then be saved for use as an initial condition in a subsequent run.

Alternatively, one can start with a known condition at sometime in the past. For example a model of cycles of ice sheet growth and decay could use a condition of no ice sheet as an initial condition, or a two-dimensional flowline model could use a profile measured 20 or 30 years ago. In the latter case, the model could be validated by comparing the final profile with one measured recently. The model might then be run into the future to predict the effects of various climate-change scenarios.

As just noted, forcing a time-dependent model usually involves varying the boundary conditions at the surface in some prescribed way. Boundary conditions at the bed or along an upstream or downstream boundary might also be varied by the modeler, but more frequently these will be calculated within the model as part of the solution. Relevant conditions at the surface are usually precipitation and temperature. These may be estimated from empirical relations, such as a relation between mean annual temperature and the Milankovitch cycles, or may be calculated in another model, such as a global climate model (GCM). If the output from an ice-sheet model is used as input for a time step in a GCM, the output of which is then used for the next time step in the ice-sheet model, the models are said to be coupled.

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