A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106984/1/fld3887.pd

Bridging Scales in 2- and 3-Dimensional Atmospheric Modeling with Adaptive Mesh Refinement

Complex multi-scale atmospheric phenomena, like tropical cyclones, challenge conventional weather and climate models, which use relatively coarse uniform-grid resolutions to cope with computational costs. Adaptive Mesh Refinement (AMR) techniques mitigate these challenges by dynamically and transiently placing high-resolution grids over salient features, thus providing sufficient local resolution while limiting the computational burden. This thesis explores the development of AMR, a technique that has been featured only sporadically in the atmospheric science literature, within a new nonhydrostatic, finite-volume dynamical core and demonstrates AMR's effectiveness in improving model accuracy and ability to resolve multi-scale features. This high-order finite-volume model implements adaptive refinement in both space and time on a cubed-sphere grid using a mapped-multiblock mesh technique developed with the Chombo AMR library. The AMR dynamical core is implemented in a hierarchy of models of increasing complexity, from an idealized 2D shallow water configuration to the nonhydrostatic 3D equation set with subgrid-scale parameterizations schemes. AMR's numerical accuracy, computational efficiency, and ability to track and resolve multifaceted and evolving features are assessed with a variety of existing and new test cases, implemented within each model iteration. Both static and dynamic refinements are analyzed to determine the strengths and weaknesses of AMR in both complex flows with small-scale features and large-scale smooth flows. The different test cases required different AMR criteria, such as vorticity, or minimum pressure based thresholds, in order to achieve the best accuracy for cost. Simulations show that the model's AMR can accurately resolve key local features in both shallow water and 3D test cases without requiring global high-resolution grids, as the adaptive grids are able to track features of interest reliably without inducing noise or visible distortions at the coarse-fine interfaces. Furthermore, the AMR grids keep degradation of the large-scale smooth flows to a minimum. 2D and 3D physics parameterizations are able to function effectively over multiple levels of refinement, though the parameterizations are sensitive to grid resolution. AMR is most effective when refinement is triggered early or when the base uniform resolution can partially resolve the features of interests. Very coarse base resolutions lead to large initial errors that cannot be overcome by AMR. However, the addition of refinement later in the simulation still results in significant improvements, especially in resolving small-scale features. The research showed that flow properties, such as strong gradients or rainbands, can be sensitive to small changes in AMR criteria. These may delay the onset of the refinement or alter the shape of the refined area, which impacts the evolution of the flow. With coarse base resolutions, the tagging criteria must therefore be uniquely tailored to capture the early growth phases of the feature of interest. A promising refinement technique is a combination of some initial refinement and AMR. The initial refinement limits error growth at the base resolution and ensures that the model can resolve the feature of interest. Overall, AMR is shown to be a powerful modeling approach that bridges the resolution gap for extreme weather events.PHDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147504/1/joferg_1.pd

A mountain-induced moist baroclinic wave test case for the dynamical cores of atmospheric general circulation models

Idealized test cases for the dynamical cores of atmospheric general circulation models are informative tools to assess the accuracy of the numerical designs and investigate the general characteristics of atmospheric motions. A new test case is introduced that is built upon a baroclinically unstable base state with an added orographic barrier. The topography is analytically prescribed and acts as a trigger of both baroclinic Rossby waves and inertia–gravity waves on a rotating, regular-sized planet. Both dry and idealized moist configurations are suggested. The latter utilizes the Kessler warm-rain precipitation scheme. The test case enhances the complexity of the existing test suite hierarchy and focuses on the impacts of two midlatitudinal mountain ridges on the circulation. Selected simulation examples from four dynamical cores are shown. These are the Spectral Element and Finite Volume dynamical cores, which are part of the National Center for Atmospheric Research (NCAR) Community Earth System Model (CESM), versions 2.1.3 and 2.2, and the Cubed-Sphere Finite Volume dynamical cores, which is new to CESM version 2.2. In addition, the Model for Prediction Across Scales (MPAS) is tested. The overall flow patterns agree well in the four dynamical cores, but the details can vary greatly. The examples highlight the broad palette of use cases for the test case and reveal physics–dynamics coupling issues.</p

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.Comment: accepted versio

A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes

Copyright © 2015 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics Vol. 290 (2015), DOI: 10.1016/j.jcp.2015.02.045A new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.Natural Environment Research Council under the “GungHo” projec

Compatible finite element spaces for geophysical fluid dynamics

Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical uid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties