Solution-adaptive moving mesh solver for geophysical flows / Christian K�uhnlein.

Doctoral Thesis Ludwig-Maximillans-Universitat Munchen, 2011.

interaction of various scales of motion. The accurate numerical representation of such flows is limited by the available number of mesh points covering the domain of interest. Numerical simulations applying uniformly distributed grid cells waste mesh points in regions of large motion scales whereas coexisting small-scale processes cannot be adequately resolved. The current thesis offers the design, implementation, and application of an adaptive moving mesh algorithm for dynamically variable spatial resolution to the numerical simulation of nonlinear geophysical flows. For this purpose, the established geophysical flow solver EULAG was modified and extended. The non-hydrostatic, anelastic equations of EULAG are rigorously implemented in time-dependent generalised coordinates. This setting enables moving mesh adaptation by solving the equations in a straightforward approach developed in this thesis. The methodological development of the new adaptive solver is divided into three tasks: (i) The flux-form Eulerian advection scheme MPDATA employed in EULAG was extended. For transport equations in conservative form, a mass conservation law enters naturally and implies a unique compatibility condition for the solution algorithm. Here, extensions of the Eulerian MPDATA integration were developed, implemented and tested to provide full compatibility with the generalised anelastic mass conservation law (GMCL) under adaptive moving meshes. (ii) A machinery performing the numerical generation of an adaptive moving curvilinear mesh was designed and implemented in EULAG. For this purpose, an auxiliary set of parabolic moving mesh partial differential equations (MMPDEs) was employed to redistribute the existing mesh cells temporally. The solutions of the MMPDEs provide the mesh coordinates and the adaptation properties of the generated moving mesh (e.g. local mesh density) are controlled by a monitor function that varies horizontally and temporally. The form of the monitor function depends inter alia in the flow state. (iii) An efficient coding of the mesh adaptation machinery was successfully incorporated into the computational framework of EULAG. For this task, the approximation of the advective contravariant mass flux in MPDATA was developed and implemented in EULAG so to minimise errors of the incompatibility with the GMCL.
The developed adaptive moving mesh solver was thoroughly investigated by simulating a number of relevant atmospheric problems. The advection of a passive tracer in a two-dimensional shear flow demonstrated the capability of the solver to automatically adapt the local resolution to the evolving small-scale filamentary structures. For this flow, the expected advantage of the mesh adaptation was achieved: the computing time (and the error) was reduced significantly by a factor of 26 (by 20%) compared to high-resolution uniform mesh computations. Another advantage of adaptive simulations is the appearance of new physical phenomena. Here, instabilities occurring at the interface of an idealised rising thermal with the ambient air could be simulated in much greater detail. The representation of the associated mixing processes is of direct relevance for simulating cumulus convection in realistic atmospheric flows. There, the process of fine-scale mixing, i.e. entrainment and detrainment, between the cloudy and the ambient air could be better resolved by mesh adaptation.
The first application of the developed adaptive mesh solver in the three-dimensional parallelised modelling framework of EULAG to an idealised baroclinic wave life cycle demonstrated the accurate representation of the synoptic-scale flow (improved statistics) and the ability to resolve coexisting mesoscale processes. Focussing the adaptation to the developing frontal zone indicated the excitation of internal gravity waves which were nearly absent in simulations applying a uniform mesh with the same number of mesh points. As before, significant savings in computing time (at least a factor of 2) compared to equivalent results of a high-resolution uniform mesh computation were achieved for the three-dimensional simulations. A cumbersome side-effect of the successful and efficient numerical simulations was the extremely time-consuming tuning of the adaptation parameters, especially of the monitor function. So far, only a very limited number of monitor functions were tested. Systematic research will yield improved specifications of the monitor function for distinct atmospheric flows. In summary, the results obtained in this thesis show the capability and potential of adaptive moving mesh methods to simulate multiscale atmospheric flows with higher numerical accuracy and a broader coverage of motion scales. However, the adaptive moving mesh method adds substantial user complexity to the modelling system EULAG.