The paper examines recent advancements in the class of Nonoscillatory Forward-in-Time (NFT) schemes that exploit the implicit LES (ILES) properties of Multidimensional Positive
Definite Advection Transport Algorithm (MPDATA). The reported developments address both global and limited area models spanning a range of atmospheric flows, from the hydrostatic regime at planetary scale, down to mesoscale and microscale where flows are inherently nonhydrostatic. All models operate on fully unstructured (and hybrid) meshes and utilize a median dual mesh finite volume discretisation. High performance computations for global flows employ a bespoke hybrid MPI-OpenMP approach and utilise the ATLAS library. Simulations across scales—from a global baroclinic instability epitomising evolution of weather systems down to stratified orographic flows rich in turbulent phenomena due to gravity-wave breaking in
dispersive media, verify the computational advancements and demonstrate the efficacy of
ILES both in regularizing large scale flows at the scale of the mesh resolution and taking a role of a subgrid-scale turbulence model in simulation of turbulent flows in the LES regime

Christian Kühnlein (7455271)

Joanna Szmelter (1250304)

Mike Gillard (1260393)

Piotr K. Smolarkiewicz (7204997)

English

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A Class of Finite-Volume Models for Atmospheric Flows Across Scales

Loughborough University Institutional Repository

A Class of Finite-Volume Models for Atmospheric Flows AcrossScales (Invited Paper)Joanna Szmelter ∗, Mike Gillard, †Wolfson School, Loughborough University, LE11 3TU, UKPiotr K. Smolarkiewicz ‡and Christian Kühnlein §European Centre for Medium Range Weather Forecasts, Reading, RG2 9AX, UKThe paper examines recent advancements in the class of Nonoscillatory Forward-in-Time(NFT) schemes that exploit the implicit LES (ILES) properties of Multidimensional PositiveDefinite Advection Transport Algorithm (MPDATA). The reported developments address bothglobal and limited area models spanning a range of atmospheric flows, from the hydrostaticregime at planetary scale, down to mesoscale and microscale where flows are inherently non-hydrostatic. All models operate on fully unstructured (and hybrid)meshes and utilize amediandualmesh finite volume discretisation. High performance computations for global flows employa bespoke hybrid MPI-OpenMP approach and utilise the ATLAS library. Simulations acrossscales—from a global baroclinic instability epitomising evolution of weather systems downto stratified orographic flows rich in turbulent phenomena due to gravity-wave breaking indispersive media, verify the computational advancements and demonstrate the efficacy ofILES both in regularizing large scale flows at the scale of the mesh resolution and taking a roleof a subgrid-scale turbulence model in simulation of turbulent flows in the LES regime.I. Methodology OutlineThe Earth atmosphere is essentially a weakly compressible, rotating and highly turbulent fluid under gravity, withscales of motion spanning about 10 decades from planetary scale of thousands of kilometers down to dissipation scalesat a fraction of a millimetre. Within the paradigm of unstructured meshes, the current paper investigates the suitabilityof ILES for predicting complex flow realisations found in challenging atmospheric problems, while addressing a rangeof aspects of numerical accuracy and efficiency. All implicit large-eddy simulations are performed using the NFTMPDATA integrators, the truncation terms of which were shown for structured grids to provide minimal artificialviscosity sufficient to regularise numerical solutions at the grid-scale and obviate the need for explicit turbulence models.Effective simulations of all-scale atmospheric flows—e.g., cloud-resolving global weather—involve semi-implicitintegration of the non-hydrostatic compressible Euler equations under gravity on a rotating sphere, while the limitedarea flows can be described by the Lipps-Hemler anelastic system of the governing PDEs. A consistent framework fordiscrete integrations of soundproof and compressible PDEs of atmospheric dynamics [1,2] is adopted in this work. Thenumerical approach employs an edge-based, finite volume discretisation within the NFT framework [3] based on theunstructured-mesh MPDATA [4,5] and advanced preconditioned non-symmetric variational (Krylov-subspace) ellipticsolvers [6]. The specificity of the Earth atmosphere—geometrically a thin shell— necessitates direct inversion in thevertical. In the Krylov solver adopted here, a bespoke deflation preconditioner removes the inherent stiffness due to theanisotropy of the terrestrial atmosphere.This flexible methodology enables integration of the governing PDEs formulated on a differential manifold withgeneralised curvilinear coordinates, but discretised using arbitrary hybrid meshes in the computational space with aco-located data arrangement for all the prognostic variables. In consequence a spherical surface with longitude-latitudecoordinates can be resolved by discretising [0, 2pi] × [−0.5pi,+0.5pi] with a variable mesh whose image in the physicalspace corresponds to a quasi-uniform resolution [7]. The capability for global modelling uses the high-performanceATLAS library [8,9] which handles generation and domain decomposition of the mesh, and the required connectivities.While Atlas is responsible for the massively-parallel distributed memory management, the performance is enhanced∗Reader, Wolfson School, Loughborough University, LE11 3TU, UK.†Research Associate, Wolfson School, Loughborough University, LE11 3TU, UK.‡Professor, European Centre for Medium Range Weather Forecasts, Reading, RG2 9AX, UK.§Research scientist, European Centre for Medium Range Weather Forecasts, Reading, RG2 9AX, UK.at the algorithmic level using a shared-memory parallelisation. This development is aimed at architectures evolvingtowards exascale.II. Numerical ResultsA. Baroclinic instabilityThe baroclinic instability benchmark [10] illustrates the performance of the approach for global flows. Thepropagation and breaking of the baroclinic wave train is shown in Figures 1 and 2, after 10 days evolution generatedfrom a weakly perturbed unstable-equilibrium initial state, consisting of the two planetary jets in mid-latitudes. Theinitially induced velocity perturbation excites the development of the instability which in time causes the wave to steepenand form overturning fronts characteristic of natural weather systems. The results are consistent with those available inthe literature [5] and illustrate the effectiveness if ILES.Fig. 1 Colour map of vertical velocity [cm/s] and isolines of potential temperature in the vertical cross section(530 N latitude).Fig. 2 Colour map of surface meridional velocity [m/s] and and isolines of potential temperature.The fully compressible Euler equations cast in the latitude-longitude of the geospherical framework [7] wereintegrated on a dual mesh created from an octahedral reduced Gaussian grid (Figure 3) with a horizontal spacing ofapprox 55 km. An atmospheric depth of 45 km was simulated employing 30 levels with the vertical spacing increasingsmoothly with height from δz = 175 m to δz = 3300 m. The reduced Gaussian grid facilitates increased spacingnear the poles and consequently alleviates time step limitations imposed by the stability conditions on the traditionalstructured lat-long grids. The treatment of periodic and pole boundary conditions follows that derived in [7].2Fig. 3 Octahedral reduced Gaussian grid representation in the computational space (left) and physical space(right). A very coarse grid is shown for clarity.B. Flow past a hillHere, the above-described approach is used in the context of the limited area model to simulate stratified flow past atwo dimensional cosine hill. The hill’s half-width L is equal to the wavelength of the vertically propagating hydrostaticmountain wave λ = 2piU/N; where U and N denote, respectively, the ambient wind and buoyancy frequency. TheFroude number Fr = U/Nh < 1, corresponding to the dimensionless height of the hill h/L > 0.25, was chosen toillustrate a strongly forced flow response. A two dimensional triangular primary mesh generated to conform to thecosine-shaped hill was used to construct a dual computational mesh with finite volumes of polygonal shapes. Figure 4shows isentropes in a developing flow for the nonlinear regime where the breaking wave is apparent in the lee.Fig. 4 Isentropes simulated using the two-dimensional limited area anelastic non-hydrostatic model; Fr < 1.III. RemarksA study demonstrating the applicability and flexibility of the finite volume discretisation operating on unstructuredmeshes for the NFT class of atmospheric nonhydrostatic models shows that the approach can complement established,semi-implicit semi-Lagrangian NWP methods. A record of consistent and accurate simulations using the advocatedapproach keeps growing [11,12] and provides the evidence that the approach has the potential to efficiently resolve abroad range of atmospheric motions, from planetary down to convection where non-hydrostatic effects dominate. Furthertechnical challenges of time to solution and energy efficiency are currently being addressed within Energy-efficientScalable Algorithms for weather Prediction at Exascale (ESCAPE) programme of research.AcknowledgmentsThis work was supported by the funding received from the European Research Council under the European Union’sSeventh Framework Programme (FP7/2012/ERCGrant agreement no. 320375), and from the ESCAPE project; ESCAPEis funded by the European Commission under the Horizon 2020 Programme - grant agreement 671627.3References[1] P.K. Smolarkiewicz, C. Kühnlein and N.P. Wedi "A consistent framework for discrete integrations of soundproofand compressible PDEs of atmospheric dynamics". J. Comput. Phys. Vol. 263, pp. 185-205, 2014.[2] P.K. Smolarkiewicz, W. Deconinck, M. Hamrud, C Kühnlein, G. Mozdzynski, J. Szmelter, N. Wedi, "Afinite-volume module for simulating global all-scale atmospheric flows". J. Comput. Phys., Vol. 314, pp. 287-304,2016.[3] J. Szmelter and P.K. Smolarkiewicz, "An edge-based unstructured mesh framework for atmospheric flows".Computers and Fluids, Vol. 46. pp. 455-460, 2010.[4] P.K. Smolarkiewicz and J. Szmelter, "MPDATA: An Edge-Based Unstructured-Grid Formulation". J. Comput.Phys. Vol. 206, pp. 624-649, 2005.[5] C. Kühnlein and P.K. Smolarkiewicz, "An unstructured-mesh finite-volume MPDATA for compressibleatmospheric dynamics".J. Comput. Phys. Vol. 334, pp. 16-30, 2017.[6] P.K. Smolarkiewicz and J. Szmelter, "A nonhydrostatic unstructured-mesh soundproof model for simulation ofinternal gravity waves, Acta Geophysica". Vol. 59, pp. 1109-1134, 2011.[7] J. Szmelter and P.K. Smolarkiewicz, "An edge-based unstructured mesh discretisation in geospherical frame-work".J. Comput. Phys., Vol. 229, pp. 4980-4995, 2010.[8]W. Deconinck, M. Hamrud, C Kühnlein, G. Mozdzynski, P.K. Smolarkiewicz, J. Szmelter, N.Wedi, "Acceleratingextreme-scale Numerical Weather Prediction". Chpt. in Parallel Processing and Applied Mathematics. Ed: RWyrzykowski, Springer International Publishing, pp. 583-593, 2015.[9] W. Deconinck, P. Bauer, M. Diamantakis, M. Hamrud, C. Kühnlein, P. Maciel, G. Mengaldo, T. Quintino, B.Raoult, P.K. Smolarkiewicz, N.P. Wedi,"Atlas : A library for numerical weather prediction and climate modelling".Computer Physics Communications, Vol. 220, pp. 188–204, 2017.[10] P.A. Ullrich, C. Jablonowski, K.A. Reed, C. Zarzycki, P.H. Lauritzen, R.D. Nair, J. Kent, A. Verlet-Banide,Dynamical core model intercomparison project (DCMIP2016) test case document,https://github.com/ClimateGlobalChange/DCMIP2016.[11] P.K. Smolarkiewicz, J. Szmelter, and F. Xiao, "Simulation of all-scale atmospheric dynamics on unstructuredmeshes". J. Comput. Phys., Vol. 322, pp. 267-287, 2016.[12] P.K. Smolarkiewicz, C. Kühnlein andW.W. Grabowski,"A finite-volume module for cloud-resolving simulationsof global atmospheric flows". J. Comput. Phys., Vol. 341, pp. 208-229, 2017.4

A class of finite-volume models for atmospheric flows across scales

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A class of finite-volume models for atmospheric flows across scales

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