Preface: multiple scales in fluid dynamics and meteorology

Abstract

Atmospheric flows and many fluid flows of engineering interest feature a multitude of characteristic length and time scales and associated scale-dependent processes. In theoretical investigations, quantitativemeteorological forecasts, or in engineering design, the ensuing complexity is often handled by employing techniques of computational fluid dynamics. Yet, in a typical application, even the most powerful computer available today would not allow us to resolve all the scales of a flow in detail. At the same time, we are mostly not interested in all these details anyway, but rather in a flow’s larger-scale features and effects. As a consequence, in practical flow simulations, we have to introduce a minimal spatiotemporal resolution as represented, e.g., by the size of a computational grid cell and by theminimal allowed time step. The effects of processes not resolved in space and time on the space-time grid are then approximately represented by “closure schemes” or “parameterizations”. For decades, practitioners in numerical weather forecasting proceeded as follows in this context: A target grid resolution was decided upon, depending on the expected available compute power. Then, subgrid scale process parameterizations were developed and implemented in an up-to-date flow solver (dynamical core) operating on the chosen grid. Finally, any free constants in the parameterizations were tuned for the entire simulation system to achieve the highest-possible weather prediction skill scores. It was common sense that any sizeable increase in grid resolution would necessarily have to be followed up by a re-tuning or even a partial rewrite of the subgrid scale process parameterizations. The situation is similar for engineering turbulent flow simulation using “large eddy simulation” approaches. In recent years, however, meteorologists as well as fluids engineers have become aware of the potential benefits of dynamically adaptive computational grids, and major meteorological centers and engineering research groups are incorporating grid adaptivity in their next-generation dynamical cores. Dynamically adaptive grids come with a caveat, though, as regards subgrid scale parameterizations. Because the spatiotemporal resolution that would be found in a simulation at any given point in space and time is not known in advance in an adaptive simulation, it is no longer possible to “tune” one’s parameterizations to perfection and then perform all production runs with the optimal parameter setting. Rather, one now needs adaptive parameterizations as well, which dynamically move particular processes from the realm of “grid resolved features” to the realm of the subgrid scales, and do so smoothly in the intermediate regimes, where the relevant processes are only partially resolved at the current resolution. Similarly, intelligent refinement and coarsening criteria are needed that place high resolution where it produces the largest benefit given some overall goal for the simulation. Needless to say that the development of such adaptive parameterizations requires to account for the subtle competition of (partially resolved) subgridscale processes with truncation errors of the underlying computational flow solver and that this necessitates balanced cooperations between physical/mathematical modellers and numerical analysts.