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Adaptive Semi-Lagrangian Modelling of Multiscaled Phenomena on Parallel Computers


Author: Dr. Jörn Behrens

Introduction

In many areas of mathematical simulation phenomena with a multitude of scales in time or space occur. These problems are difficult to handle from the mathematical point of view as well as from the computational point of view. Normally, a mathematical description of the phenomenon is valid only for a selection of scales, whereas the remaining small-scale influences are parametrized. When looking at the motion of ocean water, for example, one will observe that there is a large scale circulation, which is, however, influenced by small-scale eddy activity.

Ideas

Adaptive grid methods allow for a high local resolution of small-scale phenomena, without exhausting limited computing resources. Adaptive methods refine the grid locally and dynamically at those regions of the computational domain, where small scale phenomena are active. In order to achieve this, a local criterion for grid refinement has to be found. We are concerned with the application of adaptive methods to the simulation of ocean and atmosphere circulation. As these problems reqire large computational domains and long integration times, the simulations have to be carried out on parallel high performance computers. Therefore, Parallel adaptive grid generation is one essential part of all simulation software for multiscalemodelling.

Example

The simulation of tracer transport in the arctic stratosphere is one good example for the benefits of adaptive modelling. Figure 1 shows a typical wind situation in the arctic winter stratosphere (70 hPa layer). The polar vortex is clearly observable. Meteorologists are interrested in the simulation of the advection of a passive tracer. This high resolution simulation shows small scale filaments wich are suspected in reality (fig. 2). The simulation is feasible only due to the adaptive modelling technique. An adaptively refined grid is shown in figure 3. The tracer simulation can be watched as an amimation:

Collaboration

This project has been started at Alfred-Wegener-Institute for Polar and Marine Research (AWI), Bremerhaven and Potsdam, Germany. At the Chair for Num. Analysis and Scientific Computing it is partly continued as a collaboration with AWI. Some of the mathematical methods utilized in the simulation have been developed in cooperation with other institutions, e.g.

Figures

Figure 1 - Given wind field (initial time)
arctwind.png
Figure 2 - Locally refined grid yields filamental structure
arctloc.png
Figure 3 - In contrast to fig. 2, a coarse uniform grid reveals less structure
arctglob.png
Figure 4 - Grid corresponding to fig. 2
arctgrid.png

Animations

Some animations (quicktime movie files) show the advantage of local refinement: