Multidimensional method-of-lines transport for atmospheric flows over steep terrain using arbitrary meshes

Including terrain in atmospheric models gives rise to mesh distortions near the lower boundary that can degrade accuracy and challenge the stability of transport schemes. Multidimensional transport schemes avoid splitting errors on distorted, arbitrary meshes, and method-of-lines schemes have low computational cost because they perform reconstructions at fixed points. This paper presents a multidimensional method-of-lines finite volume transport scheme, “cubicFit”, which is designed to be numerically stable on arbitrary meshes. Stability conditions derived from a von Neumann analysis are imposed during model initialisation to obtain stability and improve accuracy in distorted regions of the mesh, and near steeply-sloping lower boundaries. Reconstruction calculations depend upon the mesh only, needing just one vector multiply per face per time-stage irrespective of the velocity field. The cubicFit scheme is evaluated using three, idealised numerical tests. The first is a variant of a standard horizontal transport test on severely distorted terrain-following meshes. The second is a new test case that assesses accuracy near the ground by transporting a tracer at the lower boundary over steep terrain on terrain-following meshes, cut-cell meshes, and new, slanted-cell meshes that do not suffer from severe time-step constraints associated with cut cells. The third, standard test deforms a tracer in a vortical flow on hexagonal-icosahedral meshes and cubed-sphere meshes. In all tests, cubicFit is stable and largely insensitive to mesh distortions, and cubicFit results are more accurate than those obtained using a multidimensional linear upwind transport scheme. The cubicFit scheme is second-order convergent regardless of mesh distortions

Numerical representation of mountains in atmospheric models

Numerical weather and climate models are using increasingly fine meshes that resolve small-scale, steeply-sloping terrain. Terrain-following meshes become highly distorted above such steep slopes, degrading the numerical balance between the pressure gradient and gravity. Furthermore, existing models often prefer dimensionally-split transport schemes for their computational efficiency, but such schemes can suffer from splitting errors above steep slopes. The cut cell method offers an alternative that avoids most mesh distortions, but arbitrarily small cut cells can impose severe time-step constraints on explicit transport schemes. This thesis makes three contributions to improve atmospheric simulations, particularly in the vicinity of steeply-sloping terrain. First, a multidimensional finite volume transport scheme is formulated to obtain accurate solutions on arbitrary, highly-distorted meshes. Stability conditions derived from a von Neumann stability analysis are imposed during model initialisation to obtain stability and improve accuracy near steeply-sloping lower boundaries. Reconstruction calculations depend upon the mesh only, needing just one vector multiply per face per time-stage. The scheme achieves second-order convergence across a series of tests using highly-distorted terrain-following meshes and cut cell meshes. The scheme is extended using the k-exact method to achieve third-order convergence on distorted meshes without increasing the computational cost during integration. Second, a new type of mesh is designed to avoid severe mesh distortions associated with terrain-following meshes and avoids severe time-step constraints associated with cut cells. Numerical experiments compare the new mesh with terrain-following and cut cell meshes, revealing that the new mesh simultaneously achieves an accurate balance between the pressure gradient and gravity, and avoids severe time-step constraints. Third, a new two-dimensional test case is proposed that excites the Lorenz computational mode. The new test is used to compare results from a nonhydrostatic model with Lorenz staggering with those from a model variant with a newly-developed generalised Charney–Phillips staggering for arbitrary meshes

Simulation of all-scale atmospheric dynamics on unstructured meshes

The advance of massively parallel computing in the nineteen nineties and beyond encouraged finer grid intervals in numerical weather-prediction models. This has improved resolution of weather systems and enhanced the accuracy of forecasts, while setting the trend for development of unified all-scale atmospheric models. This paper first outlines the historical background to a wide range of numerical methods advanced in the process. Next, the trend is illustrated with a technical review of a versatile nonoscillatory forward-in-time finite-volume (NFTFV) approach, proven effective in simulations of atmospheric flows from small-scale dynamics to global circulations and climate. The outlined approach exploits the synergy of two specific ingredients: the MPDATA methods for the simulation of fluid flows based on the sign-preserving properties of upstream differencing; and the flexible finite-volume median-dual unstructured-mesh discretisation of the spatial differential operators comprising PDEs of atmospheric dynamics. The paper consolidates the concepts leading to a family of generalised nonhydrostatic NFTFV flow solvers that include soundproof PDEs of incompressible Boussinesq, anelastic and pseudo-incompressible systems, common in large-eddy simulation of small- and meso-scale dynamics, as well as all-scale compressible Euler equations. Such a framework naturally extends predictive skills of large-eddy simulation to the global atmosphere, providing a bottom-up alternative to the reverse approach pursued in the weather-prediction models. Theoretical considerations are substantiated by calculations attesting to the versatility and efficacy of the NFTFV approach. Some prospective developments are also discussed

Unstructured mesh methods for stratified turbulent flows

Developments are reported of unstructured-mesh methods for simulating stratified, turbulent and shear flows. The numerical model employs nonoscillatory forward in-time integrators for anelastic and incompressible flow PDEs, built on Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) and a preconditioned conjugate residual elliptic solver. Finite-volume spatial discretisation adopts an edge-based data structure. Tetrahedral-based and hybrid-based median-dual options for unstructured meshes are developed, enabling flexible spatial resolution. Viscous laminar and detached eddy simulation (DES) flow solvers are developed based on the edge-based NFT MPDATA scheme. The built-in implicit large eddy simulation (ILES) capability of the NFT scheme is also employed and extended to fully unstructured tetrahedral and hybrid meshes. Challenging atmospheric and engineering problems are solved numerically to validate the model and to demonstrate its applications. The numerical problems include simulations of stratified, turbulent and shear flows past obstacles involving complex gravity-wave phenomena in the lee, critical-level laminar-turbulence transitioning and various vortex structures in the wake. Qualitative flow patterns and quantitative data analysis are both presented in the current study

Comparison of dimensionally-split and multi-dimensional atmospheric transport schemes for long time-steps

Dimensionally split advection schemes are attractive for atmospheric modelling due to their efficiency and accuracy in each spatial dimension. Accurate long time steps can be achieved without significant cost using the flux-form semi-Lagrangian technique. The dimensionally split scheme used in this paper is constructed from the one-dimensional Piecewise Parabolic Method and extended to two dimensions using COSMIC splitting. The dimensionally split scheme is compared with a genuinely multi-dimensional, method of lines scheme which, with implicit time-stepping, is stable for Courant numbers significantly larger than one. Two-dimensional advection test cases on Cartesian planes are proposed that avoid the complexities of a spherical domain or multi-panel meshes. These are solid body rotation, horizontal advection over orography and deformational flow. The test cases use distorted non-orthogonal meshes either to represent sloping terrain or to mimic the distortions near cubed-sphere edges. Mesh distortions are expected to accentuate the errors associated with dimension splitting, however, the accuracy of the dimensionally split scheme decreases only a little in the presence of mesh distortions. The dimensionally split scheme also loses some accuracy when long time-steps are used. The multi-dimensional scheme is almost entirely insensitive to mesh distortions and asymptotes to second-order accuracy at high resolution. As is expected for implicit time-stepping, phase errors occur when using long time-steps but the spatially well-resolved features are advected at the correct speed and the multi-dimensional scheme is always stable. A naive estimate of computational cost (number of multiplies) reveals that the implicit scheme is the most expensive, particularly for large Courant numbers. If the multi-dimensional scheme is used instead with explicit time-stepping, the Courant number is restricted to less than one, the accuracy is maintained and the cost becomes similar to the dimensionally split scheme

Cut Cell Methods in Global Atmospheric Dynamics

In this thesis, we study next generation techniques for the numerical simulation of global atmospheric dynamics, which range from modeling and grid generation to discretization schemes. Based on a detailed dimensional analysis of the compressible three-dimensional Navier-Stokes equations for small- and large-scale motions in the atmosphere, we derive the compressible Euler equations, the dynamical core of meteorological models. We also provide an insight into multiscale modeling and present a new numerical way of deriving reduced atmospheric models and gaining consistency of the modeling and discretization errors. The main focus of this thesis is the grid generation of the atmosphere. With regard to newly available surveys of the Earth's surface and the ever increasing computing capacities, the atmospheric triangulation techniques have to be reconsidered. In particular, the widely-used terrain-following coordinates prove to be disadvantaguous for highly resolved grids, since both the pressure gradient force error and the hydrostatic inconsistency of this vertical ansatz seriously increase with finer resolution. After a detailed analysis of the standard methods for vertical atmospheric triangulations, we present the cut cell approach as capable alternative. We construct a special cut cell method with two stabilizing constraints and provide a comprehensive guideline for an implementation of cut cells into existing atmospheric codes. For the spatial discretization of the dynamical core, we choose the Finite Volume method because of its favorable characteristics concerning conservation properties and handling of hyperbolicity. We accompany the Finite Volume discretization by a new non-linear interpolation scheme of the velocity field, which is adapted to the geometry and rotation of the Earth. To fathom the capabilities of cut cell grids together with our discretization and new interpolation scheme, we finally present several three-dimensional simulation runs. We apply standard benchmarks like an advection test and the simulation of a Rossby-Haurwitz wave and construct a new test case of counterbalancing flow between high- and low-pressure areas, with which we expose the potential of cut cell methods and the influences of different effects of the Euler equations as well as the topography of the Earth

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

Numerical weather prediction (NWP) is in a period of transition. As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where it will be necessary to use a single model for both applications. The new dynamical cores at the heart of these unified models are designed to scale efficiently on clusters with hundreds of thousands or even millions of CPU cores and GPUs. Operational and research NWP codes currently use a wide range of numerical methods: finite differences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin, which constitute element-based Galerkin (EBG) methods.Due to their important role in this transition, will EBGs be the dominant power behind NWP in the next 10 years, or will they just be one of many methods to choose from? One decade after the review of numerical methods for atmospheric modeling by Steppeler et al. (Meteorol Atmos Phys 82:287–301, 2003), this review discusses EBG methods as a viable numerical approach for the next-generation NWP models. One well-known weakness of EBG methods is the generation of unphysical oscillations in advection-dominated flows; special attention is hence devoted to dissipation-based stabilization methods. Since EBGs are geometrically flexible and allow both conforming and non-conforming meshes, as well as grid adaptivity, this review is concluded with a short overview of how mesh generation and dynamic mesh refinement are becoming as important for atmospheric modeling as they have been for engineering applications for many years.The authors would like to thank Prof. Eugenio Oñate (U. Politècnica de Catalunya) for his invitation to submit this review article. They are also thankful to Prof. Dale Durran (U. Washington), Dr. Tommaso Benacchio (Met Office), and Dr. Matias Avila (BSC-CNS) for their comments and corrections, as well as insightful discussion with Sam Watson, Consulting Software Engineer (Exa Corp.) Most of the contribution to this article by the first author stems from his Ph.D. thesis carried out at the Barcelona Supercomputing Center (BSCCNS) and Universitat Politècnica de Catalunya, Spain, supported by a BSC-CNS student grant, by Iberdrola Energías Renovables, and by grant N62909-09-1-4083 of the Office of Naval Research Global. At NPS, SM, AM, MK, and FXG were supported by the Office of Naval Research through program element PE-0602435N, the Air Force Office of Scientific Research through the Computational Mathematics program, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. The scalability studies of the atmospheric model NUMA that are presented in this paper used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. SM, MK, and AM are grateful to the National Research Council of the National Academies.Peer ReviewedPostprint (author's final draft

Electronic chip cooling system using graphite fins

International audienceAs electronic devices get smaller, cooling systems with higher thermal efficiency is demanding by fast growing electronic industry. Great amount of research has been performed on the cooling systems but research on the materials of the cooling systems needs more work. Graphite with high thermal conductivity and light weight is a great candidate to be used in electronic devices. The bottleneck of using graphene in the cooling systems is the thermal transport among the interface from the substrate to the graphene fin system. In this research finite element simulation of graphite fin cooling system has been investigated to study the effect of different applied pressure on the cooling system performance. Study of this cooling system showed good improvement in comparison with common copper fin cooling systems. Introduction

Unstructured Grid Dynamical Modeling of Planetary Atmospheres using planetMPAS: The Influence of the Rigid Lid, Computational Efficiency, and Examples of Martian and Jovian Application

We present a new planetary global circulation model, planetMPAS, based on the state-of-the-art NCAR MPAS General Circulation Model. Taking advantage of the cross compatibility between WRF and MPAS, planetMPAS includes most of the planetWRF physics parameterization schemes for terrestrial planets such as Mars and Titan. PlanetMPAS also includes a set of physics that represents radiative transfer, dry convection, moist convection and its associated microphysics for the Jovian atmosphere. We demonstrate that, despite the rigid-lid approximation, planetMPAS is suitable to simulate the climate systems in Martian and Jovian atmospheres with potential application to slow-rotating planets such as Titan. Simulations using planetMPAS show that the new model can reproduce many aspects of the observed features on Mars and Jupiter, such as the seasonal CO2 cycle, polar argon enrichment, zonal mean temperature, and qualitative dust opacity on Mars, as well as the equatorial superrotation and banded zonal wind patterns on Jupiter.Comment: Manuscript has 61 pages, 20 figures, 2 tables, submitted to Planetary and Space Scienc

Multi-Scale Modelling of Cold Regions Hydrology

Numerical computer simulations are increasingly important tools required to address both research and operational water resource issues related to the hydrological cycle. Cold region hydrological models have requirements to calculate phase change in water via consideration of the energy balance which has high spatial variability. This motivates the inclusion of explicit spatial heterogeneity and field-testable process representations in such models. However, standard techniques for spatial representation such as raster discretization can lead to prohibitively large computational costs and increased uncertainty due to increased degrees of freedom. As well, semi-distributed approaches may not sufficiently represent all the spatial variability. Further, there is uncertainty regarding which process conceptualizations are used and the degree of required complexity, motivating modelling approaches that allow testing multiple working hypotheses. This thesis considers two themes. In the first, the development of improved modelling techniques to efficiently include spatial heterogeneity, investigate warranted model complexity, and appropriate process representation in cold region models is addressed. In the second, the issues of non-linear process cascades, emergence, and compensatory behaviours in cold regions hydrological process representations is addressed. To address these themes, a new modelling framework, the Canadian Hydrological Model (CHM), is presented. Key design goals for CHM include the ability to: capture spatial heterogeneity in an efficient manner, include multiple process representations, be able to change, remove, and decouple hydrological process algorithms, work both at point and spatially distributed scales, reduce computational overhead to facilitate uncertainty analysis, scale over multiple spatial extents, and utilize a variety of boundary and initial conditions. To enable multi-scale modelling in CHM, a novel multi-objective unstructured mesh generation software *mesher* is presented. Mesher represents the landscape using a multi-scale, variable resolution surface mesh. It was found that this explicitly captured the spatial heterogeneity important for emergent behaviours and cold regions processes, and reduced the total number of computational elements by 50\% to 90\% from that of a uniform mesh. Four energy balance snowpack models of varying complexity and degree of coupling of the energy and mass budget were used to simulate SWE in a forest clearing in the Canadian Rocky Mountains. It was found that 1) a compensatory response was present in the fully coupled models’ energy and mass balance that reduced their sensitivity to errors in meteorology and albedo and 2) the weakly coupled models produced less accurate simulations and were more sensitive to errors in forcing meteorology and albedo. The results suggest that the inclusion of a fully coupled mass and energy budget improves prediction of snow accumulation and ablation, but there was little advantage by introducing a multi-layered snowpack scheme. This helps define warranted complexity model decisions for this region. Lastly, a 3-D advection-diffusion blowing snow transport and sublimation model using a finite volume method discretization via a variable resolution unstructured mesh was developed. This found that the blowing snow calculation was able to represent the spatial redistribution of SWE over a sub-arctic mountain basin when compared to detailed snow surveys and the use of the unstructured mesh provided a 62\% reduction in computational elements. Without the inclusion of blowing snow, unrealistic homogeneous snow covers were simulated which would lead to incorrect melt rates and runoff contributions. This thesis shows that there is a need to: use fully coupled energy and mass balance models in mountains terrain, capture snow-drift resolving scales in next-generation hydrological models, employ variable resolution unstructured meshes as a way to reduce computational time, and consider cascading process interactions