Exploring a Multiresolution Modeling Approach within the Shallow-Water Equations

Abstract

The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While our long-term motivation is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, we begin with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of our approach. The multiresolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine-and coarse-mesh resolution. We explore the shallow-water system with a suite of meshes ranging from quasi-uniform resolution meshes, where grid spacing is globally uniform, to highly-variable resolution meshes, where grid spacing varies by a factor of 16 between the fine and coarse regions. We find that potential vorticity is conserved to within machine precision and total available energy is conserved to within time-truncation error. This finding holds for the full suite of meshes, ranging from quasi-uniform resolution and highly-variable resolution meshes. Using shallow-water test cases 2 and 5, we find that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This finding is consistent with results obtained by others.When these variable resolution meshes are used for the simulation of an unstable zonal jet, we find that the core features of the growing instability are largely unchanged as the variation in mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multi-resolution climate system modeling.1

A high-fidelity multiresolution digital elevation model for Earth systems

“…With the success of Earth and environment systems with these scale-diversified processes, persistent demands exist for extending their utility to new and expanding scopes (Ringler et al, 2008;Tarolli, 2014;Wilson, 2012), as exemplified by lapse-rate-controlled functional plant distributions (Ke et al, 2012), orographic forcing imposed on oceanic and atmospheric dynamics (Nunalee et al, 2015;Brioude et al, 2012;Hughes et al, 2015), topographic dominated flood inundations (Bilskie et al, 2015;Hunter et al, 2007), and many other geomorphological (Wilson, 2012), soil (Florinsky and Pankratov, 2015), and ecological (Leempoel et al, 2015) examples from Earth systems. However, as numerical simulation systems evolved to incorporate broader scales and finer processes to produce more exact predictions (Ringler et al, 2011;Weller et al, 2016;Wilson, 2012;Zarzycki et al, 2014), how to accurately assimilate or transform the finePublished by Copernicus Publications on behalf of the European Geosciences Union. 240 X. Duan et al: A high-fidelity multiresolution DEM for Earth systems resolution topography has proven to be a quite difficult task (Bilskie et al, 2015;Chen et al, 2015;Tarolli, 2014).…”

“…The sub-grid schemes are designed for the empirical parameterisation rather than accurate topography representation, which often leads to mixed-up uncertainties and bias of endogenous variability (Jiménez and Dudhia, 2013;Nunalee et al, 2015). However, under-resolved representation could be improved by variable-resolution enhancement, and bias of simulations can be justified by more fidelity topography transformation (Nunalee et al, 2015;Ringler et al, 2011). Topography is also commonly treated as a static boundary layer in dynamics simulations, where different interpolation strategies and mesh refinement techniques are used to convey terrain variation (Guba et al, 2014;Kesserwani and Liang, 2012;Nikolos and Delis, 2009;Weller et al, 2016).…”

“…But a mesh constructed from interpolated vertices does not necessarily comply with the terrain relief, and elevation errors are frequently reported as an input uncertainty (Bilskie and Hagen, 2013;Hunter et al, 2007;Nunalee et al, 2015;Wilson and Gallant, 2000). Although there are many situations where dynamic conditions are stressed as stronger impacts on predictions (Cea and Bladé, 2015;Budd et al, 2015), the underlying topography is still very important due to its increasingly improved fidelity to the Earth's surface (Bates, 2012;Tarolli, 2014), and a sophisticated topography transformation would be beneficial to reduce discrepancies arising from physical inconsistencies (Chen et al, 2015;Glover, 1999;Ringler et al, 2011).…”

“…Multiresolution is an effective paradigm to model scale diversity (Du et al, 2010;Guba et al, 2014;Ringler et al, 2011;Weller et al, 2016). Amongst a number of promising approaches, we are especially fascinated by the centroidal Voronoi tessellations (CVTs) method as an intuitive way to redistribute samples with a designated function (Du et al, 1999(Du et al, , 2010Ringler et al, 2011) to develop an optimised surface transformation method to realise multiresolution terrain models.…”

Towards convection-resolving, global atmospheric simulations with the Model for Prediction Across Scales (MPAS) v3.1: an extreme scaling experiment

“…This, however, is prohibitively expensive or simply not feasible, even on the latest generations of supercomputers. An intermediate approach therefore is to run a global model at a moderate resolution and use a smooth mesh transition on a variable-resolution grid, where filters are efficient at the local scale of the corresponding grid cell (Ringler et al, 2011). Beside the here-discussed Model for Prediction Across Scales (MPAS) 1 model, few other recent developments such as ICON (icosahedral non-hydrostatic model; Zaengel et al, 2015) adopt this strategy.…”