Embedding/nesting of models

Model embedding makes use of two numerical models; one applied to a sub-domain of the other. This sub-domain is the region of study. The larger domain is included in the modeling study because the influence of the region is perceived to be important to the evolution of the flow in the smaller domain. Typically, the large-domain model is run at lower resolution or may have simplified physics. Within the sub-domain, the resolution is chosen to resolve the phenomena of interest. The primary motivation behind the embedding approach is to reduce the computational cost. A recent study made use of this approach to study eddies in the Atlantic (Spall, 1991). The results of global scale models might also be used analogously, though most mathematicians probably regard embedded models as interactive.

To date, model embedding has been limited to physical models and our discussion reflects this fact. However, the biological analogs are obvious. For the purposes of Southern Ocean modeling, there are a few cases in which an embedding approach might be of use. One is in a study of the eddy dynamics of the ACC and the other a study of Weddell Sea processes.

Consider first the case of eddy dynamics in the ACC. To provide the large scale flow, a 0.25° (roughly 30 km) primitive equation model might be used. Forced by winds, and perhaps by surface heat flux, a passable representation of the ACC is possible. This would then provide an eddy-resolving regional model with 0.025° resolution (3 km) with the boundary conditions representative of the ACC. This model could be 300 km on a side. The large-domain model would have 122,500 grid points and the regional model would have 10,000 grid points. Assuming that both models have the same vertical discretization, the computational costs would be roughly evenly divided between each component. Resolving the entire Southern Ocean at 3 km would be 1000 times the computational cost of the large-scale model or 500 times the cost of the embedded-model strategy. Clearly, this is the difference between a numerical experiment that can be conducted and one that will not be possible for many years.

This large advantage comes at a cost. The fundamental limitation of the embedding strategy is the transfer of eddy energy between the large scale and the regional models. For this example problem, the focus of the regional model is on the eddy processes and associated eddy fluxes. A 3 km resolution model is chosen because we are interested in processes occurring at 30 km and above (roughly 10x the resolution of the model). As the flow enters the regional model at the western end of the domain, there is little or no energy at the eddy scale since the large scale model represent flows at 300 km and larger. The eddy field must be generated within the regional model as the flow carries energy from west to east. This adjustment occurs over a region that may be estimated as the product of the flow velocity and the eddy growth period; 20 cm/s * 5 days = 125 km. This is only an order of magnitude estimate and a range of 60 to 1250 km might provide reasonable bounds on the potential adjustment domain. Using the 125 km estimate, much of our domain (42%) will be taken up by an adjustment of the eddy field. Furthermore, the climatology of the ACC may be different in each model either in structure or position. These physical processes can result in significant eddy diffusion and vertical transport and therefore may be important to chemical and biological processes as well. Careful numerical experiment design will be required to separate the effects of adjustment within the regional model from the signal of ACC eddy dynamics.

A modeling study of the Weddell Sea and the formation of Antarctic water masses might be carried out with the large scale model providing conditions at the outer edge of the sea. A small scale model including nonlinear effects in the equation of state might be used to look at the processes that lead to very cold, salty water leaving the Weddell Sea. Here, the boundary condition provided by the large-scale model will not have as rapid effect upon the interior. In fact, the essence of the study would be to understand the way in which the boundary condition affects the Weddell Sea. We do not expect that the lack of eddies in the large scale model will have a dramatic effect upon the result, primarily because these eddies would not extend onto the shelf to a significant extent.

In summary, model embedding can provide an economical approach to studying systems at high resolution. Some problems are more amenable to this approach than others. Naive use of the boundary conditions from a large scale model will contaminate the regional model results with adjustment processes. Careful numerical experiment design may permit the application of this method to a greater range of problems.

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