Computational modes and grid imprinting on five quasi-uniform spherical C-grids

Currently, most operational forecasting models use latitude-longitude grids, whose convergence of meridians towards the poles limits parallel scaling. Quasi-uniform grids might avoid this limitation. Thuburn et al, JCP, 2009 and Ringler et al, JCP, 2010 have developed a method for arbitrarily-structured, orthogonal C-grids (TRiSK), which has many of the desirable properties of the C-grid on latitude-longitude grids but which works on a variety of quasi-uniform grids. Here, five quasi-uniform, orthogonal grids of the sphere are investigated using TRiSK to solve the shallow-water equations. We demonstrate some of the advantages and disadvantages of the hexagonal and triangular icosahedra, a Voronoi-ised cubed sphere, a Voronoi-ised skipped latitude-longitude grid and a grid of kites in comparison to a full latitude-longitude grid. We will show that the hexagonal-icosahedron gives the most accurate results (for least computational cost). All of the grids suffer from spurious computational modes; this is especially true of the kite grid, despite it having exactly twice as many velocity degrees of freedom as height degrees of freedom. However, the computational modes are easiest to control on the hexagonal icosahedron since they consist of vorticity oscillations on the dual grid which can be controlled using a diffusive advection scheme for potential vorticity

The seasonal cycle of gravity wave drag in the middle atmosphere

Using a variational technique, middle atmosphere gravity wave drag (GWD) is estimated from Met Office middle atmosphere analyses for the year 2002. The technique employs an adjoint model of a middle atmosphere dynamical model to minimize a cost function that measures the differences between the model state and observations. The control variables are solely the horizontal components of GWD; therefore, the minimization determines the optimal estimate of the drag. For each month, Met Office analyses are taken as the initial condition for the first day of the month, and also as observations for each successive day. In this way a three-dimensional GWD field is obtained for the entire year with a temporal resolution of 1 day. GWD shows a pronounced seasonal cycle. During solstices, there are deceleration regions of the polar jet centered at about 63° latitude in the winter hemisphere, with a peak of 49 m s-1 day-1 at 0.24 hPa in the Southern Hemisphere; the summer hemisphere also shows a deceleration region but much weaker, with a peak of 24 m s-1 day-1 centered at 45° latitude and 0.6 hPa. During equinoxes GWD is weak and exhibits a smooth transition between the winter and summer situation. The height and latitude of the deceleration center in both winter and summer hemispheres appear to be constant. Important longitudinal dependencies in GWD are found that are related to planetary wave activity; GWD intensifies in the exit region of jet streaks. In the lower tropical stratosphere, the estimated GWD shows a westward GWD descending together with the westward phase of the quasi-biennial oscillation. Above, GWD exhibits a semiannual pattern that is approximately out of phase with the semiannual oscillation in the zonal wind. Furthermore, a descending GWD pattern is found at those heights, similar in magnitude and sign to that in the lower stratosphere.Fil: Pulido, Manuel Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste. Instituto de Modelado e Innovación Tecnológica. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas Naturales y Agrimensura. Instituto de Modelado e Innovación Tecnologica; ArgentinaFil: Thuburn, John. University of Exeter. School of Engineering, Computing Science and Mathematics; Reino Unid

Numerical representation of geostrophic modes on arbitrarily structured C-grids

Copyright © 2009 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. 228, Issue 22 (2009), DOI: 10.1016/j.jcp.2009.08.006A C-grid staggering, in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces (or edges in two dimensions) is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes. However, the discretization of the Coriolis terms is non-trivial. For constant Coriolis parameter, the linearized shallow water equations support geostrophic modes: stationary solutions in geostrophic balance. A naive discretization of the Coriolis terms can cause geostrophic modes to become non-stationary, causing unphysical behaviour of numerical solutions. Recent work has shown how to discretize the Coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the Coriolis terms remain energy conserving. In this paper this result is extended to arbitrarily structured C-grids. An explicit formula is given for constructing an appropriate discretization of the Coriolis terms. The general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal C-grid and the spherical longitude–latitude C-grid. Numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal–pentagonal and triangular geodesic C-grids on the sphere

On the form of the viscous term for two dimensional Navier-Stokes flows

This is a pre-copyedited, author-produced PDF of an article accepted for publication in Quarterly Journal of Mechanics and Applied Mathematics following peer review. The version of record, Andrew D. Gilbert, Xavier Riedinger, and John Thuburn, On the form of the viscous term for two dimensional Navier–Stokes flows, Q J Mechanics Appl Math (2014) 67 (2): 205-228 first published online February 27, 2014 is available online at: http://qjmam.oxfordjournals.org/content/67/2/205The form of the viscous term is discussed for incompressible flow on a two-dimensional curved surface S and for the shallow water equations. In the case of flow on a surface three versions are considered. These correspond to taking curl twice, to applying the Laplacian defined in terms of a metric, and to taking the divergence of a symmetric stress tensor. These differ on a curved surface, for example a sphere. The three terms are related and their properties discussed, in particular energy and angular momentum conservation. In the case of the shallow water equations again three forms of dissipation are considered, the last of which involves the divergence of a stress tensor. Their properties are discussed, including energy conservation and whether the rotating bucket solution of the three-dimensional Navier–Stokes equation is reproduced. A derivation of the viscous term is also given based on shallow water equations as a truncation of the Navier–Stokes equation, with forces on a column determined by integration over the vertical. For both incompressible flow on a surface and for the shallow water equations, it is argued that a viscous term based on a symmetric stress tensor should be used as this leads to correct treatment of angular momentum

A semi-implicit version of the MPAS-atmosphere dynamical core

An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge-Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank-Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%-20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.UK Natural Environment Research Council as part of the G8 ICOMEX projec

A Lagrangian vertical coordinate version of the ENDGame dynamical core. Part II: Evaluation of Lagrangian conservation properties.

This is the author accepted manuscript.Final version available from Wiley via the DOI in this record.A baroclinic instability test case is used to compare the Lagrangian conservation properties of three versions of a semi‐implicit semi‐Lagrangian dynamical core: one using a height based vertical coordinate and two using a Lagrangian vertical coordinate. The Lagrangian coordinate versions differ in the choice of target levels to which model levels are reset after each step—the first uses the initial model level heights while the second uses quasi‐Lagrangian target levels. A range of diagnostics related to Lagrangian conservation are computed, including global entropy, unavailable energy, cross‐isentrope mass flux, and consistency of potential temperature and potential vorticity with passive tracers and parcel trajectories. The global entropy, unavailable energy, and cross‐isentrope fluxes do not suggest any clear advantage or disadvantage from the use of a Lagrangian vertical coordinate, though the cross‐isentrope flux reveals a flaw in the formulation of the remapping of potential temperature in the Lagrangian coordinate model at the top boundary. The use of a Lagrangian vertical coordinate with quasi‐Lagrangian target levels improves the consistency among potential temperature as a dynamical variable, potential temperature as a tracer and potential temperature on Lagrangian particle trajectories. It also improves consistency between a potential vorticity tracer and potential vorticity on Lagrangian particle trajectories. However, it degrades the consistency between model and tracer potential vorticity, as well as between model potential vorticity and potential vorticity on Lagrangian trajectories. This degradation appears to be related to the slopes of model levels, which are greater in the version with quasi‐Lagrangian target levels.This work was funded by the Natural Environment Research Council under grant NE/H006834/

A Lagrangian vertical coordinate version of the ENDGame dynamical core. Part I: Formulation, remapping strategies, and robustness

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this record.Previous work provides evidence that Lagrangian conservation and related properties of a numerical model dynamical core can be improved by the use of a Lagrangian or quasi-Lagrangian vertical coordinate (LVC). Most previous model developments based on this idea have made the hydrostatic approximation. Here the LVC is implemented in a nonhydrostatic compressible Euler equation dynamical core using almost identical numerical methods to ENDGame, the operational dynamical core of the Met Office atmospheric Unified Model. This enables a clean comparison of LVCand height-coordinate versions of the dynamical core using numerical methods that are as similar as possible. Since Lagrangian surfaces distort over time, model level heights are continually reset to certain ‘target levels’ and the values of model fields are remapped onto their new locations. Different choices for these target levels are discussed, along with remapping strategies that focus on different conservation or balance properties. Sample results from a baroclinic instability test case are presented. The LVC formulation is found to be rather less robust than the height-coordinate version; some reasons for this are discussed.We are grateful to Nigel Wood for pointing out the computational mode of the LVC vertical discretization. We also thank two anonymous reviewers for their constructive comments on an earlier version of this paper. This work was funded by the Natural Environment Research Council under grant NE/H006834/1

Using the UM dynamical cores to reproduce idealised 3D flows

We demonstrate that both the current (New Dynamics), and next generation (ENDGame) dynamical cores of the UK Met Office global circulation model, the UM, reproduce consistently, the long-term, large-scale flows found in several published idealised tests. The cases presented are the Held-Suarez test, a simplified model of Earth (including a stratosphere), and a hypothetical tidally locked Earth. Furthermore, we show that using simplifications to the dynamical equations, which are expected to be justified for the physical domains and flow regimes we have studied, and which are supported by the ENDGame dynamical core, also produces matching long-term, large-scale flows. Finally, we present evidence for differences in the detail of the planetary flows and circulations resulting from improvements in the ENDGame formulation over New Dynamics.Comment: 34 Pages, 23 Figures. Accepted for publication in Geoscientific Model Development (pre-proof version

A two-fluid single-column model of the dry, shear-free, convective boundary layer

This is the final version. Available on open access from Wiley via the DOI in this record.A single-column model of the dry, shear-free, convective boundary layer is presented in which nonlocal transports by coherent structures such as thermals are represented by the partitioning of the fluid into two components, updraft and environment, each with a full set of prognostic dynamical equations. Local eddy diffusive transport and entrainment and detrainment are represented by parameterizations similar to those used in Eddy Diffusivity Mass Flux schemes. The inclusion of vertical diffusion of the vertical velocity is shown to be important for suppressing an instability inherent in the governing equations. A semi-implicit semi-Lagrangian numerical solution method is presented and shown to be stable for large acoustic and diffusive Courant numbers, though it becomes unstable for large advective Courant numbers. The solutions are able to capture key physical features of the dry convective boundary layer. Some of the numerical challenges posed by sharp features in the solution are discussed, and areas where the model could be improved are highlighted.Natural Environment Research Council (NERC