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The representation of cumulus convection is a known source of uncertainty within current weather and climate models. Where model resolution is too coarse to accurately resolve convection, parameterisations are required to estimate the impact of small-scale convective processes. High resolution large eddy simulations (LES) can be used to diagnose many aspects of convective processes, such as heat and momentum budgets and rates of entrainment. However, LES is computationally expensive, making it impossible to use within operational models. This study aims to bridge the gap between current coarser models and LES by developing a stochastic Lagrangian model to represent an ensemble of air parcels. Vertical velocity, liquid water potential temperature, and total specific humidity are predicted following the ensemble of parcels. The random motions associated with turbulence are represented by a stochastic term within the w-tendency equation. The mean fields which the parcels interact with are defined by an ensemble average of nearby parcels. Several fixers have been developed to ensure that conservation properties are respected. At the current stage of development, the model can represent dry convective boundary layer and shallow convection cases. A theoretical study of the stochastic differential equations is useful to verify the self-consistency of the model and also as a tool for calibrating various parameters within the model. A key question for this project is how well the stochastic parcel model can replicate the statistics of LES results. This will act as a measure of the model’s success, allowing for a deeper understanding on accurately modelling convective processes. Due to the Lagrangian nature of the model, analysis can be conducted upon how the parcels’ characteristics change over time as the parcels experience smaller-scale convective processes such as entrainment. Ultimately, results from this model may yield better understandings of small-scale convective processes. This can create potential for improvements to parameterisations in operational models, reducing model uncertainty generated by convective processes.

The Charney–Phillips grid, used in many numerical models of the atmosphere, involves vertically staggering the nodes of the density variable with the nodes of the entropy-type variable. When moisture is included in such a model, it is either co-located with density so that moisture can be transported conservatively and consistently with dry mass, or with the entropy-type variable so that the coupling between moisture and temperature can be represented well. Both properties are desirable, yet at first it appears difficult to obtain both simultaneously. Here, we present a framework to resolve this problem, by co-locating the moisture mixing ratio with potential temperature but formulating its transport as that of a density on a vertically shifted mesh. Within this framework, particular choices of the operators involved provide the desired conservation and consistency properties of the moisture transport. The framework is described in the context of a finite-element approach. We also present an explicit Runge–Kutta time-stepping scheme that is appropriate for use within this framework. This approach is then illustrated through numerical tests, which demonstrate that it does indeed have the desired conservation and consistency properties.

We hypothesize that the convective atmospheric boundary layer is marginally stable when the damping effects of turbulence are taken into account. If the effects of turbulence are modeled as an eddy viscosity and diffusivity, then an idealized analysis based on the hypothesis predicts a well-known scaling for the magnitude of the eddy viscosity and diffusivity. It also predicts that the marginally stable modes should have vertical and horizontal scales comparable to the boundary layer depth. A more quantitative numerical linear stability analysis is presented for a realistic convective boundary layer potential temperature profile and is found to support the hypothesis.

Approximations in the moist thermodynamics of atmospheric/weather models are often inconsistent. Different parts of numerical models may handle the thermodynamics in different ways, or the approximations may disagree with the laws of thermodynamics. In order to address these problems we may derive all relevant thermodynamic quantities from a defined thermodynamic potential; approximations are then instead made to the potential itself --- this guarantees self-consistency. This concept is viable for vapor and liquid water mixtures in a moist atmospheric system using the Gibbs function but on extension to include the ice phase an ambiguity exists at the triple-point. In order to resolve this the energy function must be used instead; constrained maximisation methods may be used on the entropy in order to solve the system equilibrium state. Once this is done however, a further extension is necessary for atmospheric systems. In the Earth's atmosphere many important non-equilibrium processes take place; for example, freezing of super-cooled water, evaporation, and precipitation. To fully capture these processes the equilibrium method must be reformulated to involve finite rates of approach towards equilibrium. This may be done using various principles of non-equilibrium thermodynamics, principally Onsager reciprocal relations. A numerical scheme may then be implemented which models the competing finite processes in a moist thermodynamic system.

Dealing with extreme scale earth system models is challenging from the computer science perspective, as the required computing power and storage capacity are steadily increasing. Scientists perform runs with growing resolution or aggregate results from many similar smaller-scale runs with slightly different initial conditions (the so-called ensemble runs). In the fifth Coupled Model Intercomparison Project (CMIP5), the produced datasets require more than three Petabytes of storage and the compute and storage requirements are increasing significantly for CMIP6. Climate scientists across the globe are developing next-generation models based on improved numerical formulation leading to grids that are discretized in alternative forms such as an icosahedral (geodesic) grid. The developers of these models face similar problems in scaling, maintaining and optimizing code. Performance portability and the maintainability of code are key concerns of scientists as, compared to industry projects, model code is continuously revised and extended to incorporate further levels of detail. This leads to a rapidly growing code base that is rarely refactored. However, code modernization is important to maintain productivity of the scientist working with the code and for utilizing performance provided by modern and future architectures. The need for performance optimization is motivated by the evolution of the parallel architecture landscape from homogeneous flat machines to heterogeneous combinations of processors with deep memory hierarchy. Notably, the rise of many-core, throughput-oriented accelerators, such as GPUs, requires non-trivial code changes at minimum and, even worse, may necessitate a substantial rewrite of the existing codebase. At the same time, the code complexity increases the difficulty for computer scientists and vendors to understand and optimize the code for a given system. Storing the products of climate predictions requires a large storage and archival system which is expensive. Often, scientists restrict the number of scientific variables and write interval to keep the costs balanced. Compression algorithms can reduce the costs significantly but can also increase the scientific yield of simulation runs. In the AIMES project, we addressed the key issues of programmability, computational efficiency and I/O limitations that are common in next-generation icosahedral earth-system models. The project focused on the separation of concerns between domain scientist, computational scientists, and computer scientists. The key outcomes of the project described in this article are the design of a model-independent Domain-Specific Language (DSL) to formulate scientific codes that can then be mapped to architecture specific code and the integration of a compression library for lossy compression schemes that allow scientists to specify the acceptable level of loss in precision according to various metrics. Additional research covered the exploration of third-party DSL solutions and the development of joint benchmarks (mini-applications) that represent the icosahedral models. The resulting prototypes were run on several architectures at different data centers.

Context. Atmospheric superrotating flows at the equator are an almost ubiquitous result of simulations of hot Jupiters, and a theory explaining how this zonally coherent flow reaches an equilibrium has been developed in the literature. However, this understanding relies on the existence of either an initial superrotating or a sheared flow, coupled with a slow evolution such that a linear steady state can be reached. Aims. A consistent physical understanding of superrotation is needed for arbitrary drag and radiative timescales, and the relevance of considering linear steady states needs to be assessed. Methods. We obtain an analytical expression for the structure, frequency and decay rate of propagating waves in hot Jupiter atmospheres around a state at rest in the 2D shallow water beta plane limit. We solve this expression numerically and confirm the robustness of our results with a 3D linear wave algorithm. We then compare with 3D simulations of hot Jupiter atmospheres and study the non linear momentum fluxes. Results. We show that under strong day night heating the dynamics does not transit through a linear steady state when starting from an initial atmosphere in solid body rotation. We further show that non linear effects favour the initial spin up of superrotation and that the acceleration due to the vertical component of the eddy momentum flux is critical to the initial development of superrotation. Conclusions. Overall, we describe the initial phases of the acceleration of superrotation, including consideration of differing radiative and drag timescales, and conclude that eddy-momentum driven superrotating equatorial jets are robust, physical phenomena in simulations of hot Jupiter atmospheres., Comment: 28 pages, 20 pages of text - 8 of appendices, 9 figures in text - 6 in appendices

We are grateful to 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/N013123/1 as part of the ParaCon programme.

A new theoretical framework is derived for parameterization of subgrid physical processes in atmospheric models; the application to parameterization of convection and boundary layer fluxes is a particular focus. The derivation is based on conditional filtering, which uses a set of quasi-Lagrangian labels to pick out different regions of the fluid, such as convective updrafts and environment, before applying a spatial filter. This results in a set of coupled prognostic equations for the different fluid components, including subfilter-scale flux terms and entrainment/detrainment terms. The framework can accommodate different types of approaches to parameterization, such as local turbulence approaches and mass flux approaches. It provides a natural way to distinguish between local and nonlocal transport processes and makes a clearer conceptual link to schemes based on coherent structures such as convective plumes or thermals than the straightforward application of a filter without the quasi-Lagrangian labels. The framework should facilitate the unification of different approaches to parameterization by highlighting the different approximations made and by helping to ensure that budgets of energy, entropy, and momentum are handled consistently and without double counting. The framework also points to various ways in which traditional parameterizations might be extended, for example, by including additional prognostic variables. One possibility is to allow the large-scale dynamics of all the fluid components to be handled by the dynamical core. This has the potential to improve several aspects of convection–dynamics coupling, such as dynamical memory, the location of compensating subsidence, and the propagation of convection to neighboring grid columns.

The erosion of a model stratospheric polar vortex in response to bottom boundary forcing is investigated numerically. Stripping of filaments of air from the polar vortex has been implicated in the occurrence of stratospheric sudden warmings (SSWs) but it is not understood in detail what factors determine the rate and amount of stripping. Here a shallow water vortex forced by topography is used to investigate the factors initiating stripping and whether this leads the vortex to undergo an SSW. It is found that the amplitude of topographic forcing must exceed some threshold (of order 200–450 m) in order for significant stripping to occur. For larger forcing amplitudes significant stripping occurs, but not as an instantaneous response to the forcing; rather, the forcing appears to initiate a process that ultimately results in stripping several tens of days later. There appears to be no simple quantitative relationship between the amount of mass stripped and the topography amplitude. However, at least over the early stages of the experiments, there is a good correlation between the amount of mass stripped and the global integral of wave activity, which may be interpreted as a measure of the accumulated topographic forcing. Finally there does not appear to be a simple correspondence between amount of mass stripped and the occurrence of an SSW.

The thermodynamics of moist processes is complicated, and in typical atmospheric models numerous approximations are made. However, they are not always made in a self-consistent way, which could lead to spurious sources or sinks of energy and entropy. One way to ensure self-consistency is to derive all thermodynamic quantities from a thermodynamic potential such as the Gibbs function. Approximations may be made to the Gibbs function; these approximations are inherited by all derived quantities in a way that guarantees self-consistency. Here, the feasibility of using the Gibbs function in an atmospheric model is demonstrated through the development of a semi-implicit, semi-Lagrangian vertical slice model, and its application to a standard buoyant bubble test case. The flexibility of the approach is also demonstrated by running the test case with four different equations of state corresponding to dry air, moist air that is saturated, a pseudo-incompressible fluid, and an incompressible fluid. A recently presented ‘blended’ equation set that unifies the dry fully compressible case and the pseudo-incompressible case is also easily accommodated.

To meet the challenges posed by future generations of massively parallel supercomputers a reformulation of the dynamical core for the Met Office’s weather and climate model is presented. This new dynamical core uses explicit finite‐volume type discretisations for the transport of scalar fields coupled with an iterated‐implicit, mixed finite‐element discretisation for all other terms. The target model aims to maintain the accuracy, stability and mimetic properties of the existing Met Office model independent of the chosen mesh while improving the conservation properties of the model. This paper details that proposed formulation and, as a first step towards complete testing, demonstrates its performance for a number of test cases in (the context of) a Cartesian domain. The new model is shown to produce similar results to both the existing semi‐implicit semi‐Lagrangian model used at the Met Office and other models in the literature on a range of bubble tests and orographically forced flows in two and three dimensions.

Context. The study of linear waves and instabilities is necessary to understand the physical evolution of an atmosphere, and can provide physical interpretation of the complex flows found in simulations performed using Global Circulation Models (GCM). In particular, the acceleration of superrotating flow at the equator of hot Jupiters has mostly been studied under several simplifying assumptions, the relaxing of which may impact final results. Aims. We develop and benchmark a publicly available algorithm to identify the eigenmodes of an atmosphere around any initial steady state. We also solve for linear steady states. Methods. We linearise the hydrodynamical equations of a planetary atmosphere in a steady state with arbitrary velocities and thermal profile. We then discretise the linearised equations on an appropriate staggered grid, and solve for eigenvectors and linear steady solutions with the use of a parallel library for linear algebra: ScaLAPACK. We also implement a posteriori calculation of an energy equation in order to obtain more information on the underlying physics of the mode. Results. Our code is benchmarked against classical wave and instability test cases in multiple geometries. The steady linear circulation calculations also reproduce expected results for the atmosphere of hot Jupiters. We finally show the robustness of our energy equation, and its power to obtain physical insight into the modes. Conclusions. We have developed and benchmarked a code for the study of linear processes in planetary atmospheres, with an arbitrary steady state. The calculation of an a posteriori energy equation provides both increased robustness and physical meaning to the obtained eigenmodes. This code can be applied to various problems, and notably to further study the initial spin up of superrotation of GCM simulations of hot Jupiter., Comment: 18 pages (6 of appendix), 9 figures

The linear stability of two well known energy and enstrophy conserving schemes for the vector invariant hydrostatic primitive equations is examined. The problem is analysed for a stably stratified Boussinesq fluid on an f-plane, with a constant velocity field, in height and isopycnal coordinates, by separation of variables into vertical normal modes and a linearised form of the shallow water equations (SWEs). As found by [Hollingsworth ~ al.(1983)Hollingsworth, Kallberg, Renner and Burridge], (HKRB hereafter) the schemes are linearly unstable in height coordinate models, due to a non-cancellation of terms in the momentum equations. The schemes with the modified formulations of the kinetic energy proposed by HKRB are shown to have Hermitain stability matrices and hence to be stable to all perturbations. All perturbations in isopycnal models are also shown to be neutrally stable, even with the original formulations for kinetic energy. Analytical expressions are derived for the smallest equivalent depths obtained using Charney-Phillips and Lorenz vertical grids, which show that the Lorenz grid has larger growth rates for the unstable schemes than the Charney-Phillips grid. Test cases are proposed for assessing the stability of new numerical schemes using the SWEs.

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.

A warm bias in tropical tropopause temperature is found in the Met Office Unified Model (MetUM), in common with most models from phase 5 of CMIP (CMIP5). Key dynamical, microphysical, and radiative processes influencing the tropical tropopause temperature and lower-stratospheric water vapor concentrations in climate models are investigated using the MetUM. A series of sensitivity experiments are run to separate the effects of vertical advection, ice optical and microphysical properties, convection, cirrus clouds, and atmospheric composition on simulated tropopause temperature and lower-stratospheric water vapor concentrations in the tropics. The numerical accuracy of the vertical advection, determined in the MetUM by the choice of interpolation and conservation schemes used, is found to be particularly important. Microphysical and radiative processes are found to influence stratospheric water vapor both through modifying the tropical tropopause temperature and through modifying upper-tropospheric water vapor concentrations, allowing more water vapor to be advected into the stratosphere. The representation of any of the processes discussed can act to significantly reduce biases in tropical tropopause temperature and stratospheric water vapor in a physical way, thereby improving climate simulations.

We present highlights from a large set of simulations of a hot Jupiter atmosphere, nominally based on HD 209458b, aimed at exploring both the evolution of the deep atmosphere, and the acceleration of the zonal flow or jet. We find the occurrence of a super-rotating equatorial jet is robust to changes in various parameters, and over long timescales, even in the absence of strong inner or bottom boundary drag. This jet is diminished in one simulation only, where we strongly force the deep atmosphere equator-to-pole temperature gradient over long timescales. Finally, although the eddy momentum fluxes in our atmosphere show similarities with the proposed mechanism for accelerating jets on tidally-locked planets, the picture appears more complex. We present tentative evidence for a jet driven by a combination of eddy momentum transport and mean flow., 26 pages, 22 Figures. Accepted for publication in Astronomy and Astrophysics

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/67/2/205

Today’s global Earth system models began as simple regional models of tropospheric weather systems. Over the past century, the physical realism of the models has steadily increased, while the scope of the models has broadened to include the global troposphere and stratosphere, the ocean, the vegetated land surface, and terrestrial ice sheets. This chapter gives an approximately chronological account of the many and profound conceptual and technological advances that made today’s models possible. For brevity, we omit any discussion of the roles of chemistry and biogeochemistry, and terrestrial ice sheets.

The equations governing atmospheric flow imply transfers of energy and potential enstrophy between scales. Accurate simulation of turbulent flow requires that numerical models, which have finite resolution and truncation errors, adequately capture these interscale transfers, particularly between resolved and unresolved scales. It is therefore important to understand how accurately these transfers are modelled in the presence of scale-selective dissipation or other forms of subgrid model. Here, the energy and enstrophy cascades in numerical models of two-dimensional turbulence are investigated using the barotropic vorticity equation. Energy and enstrophy transfers in spectral space due to truncated scales are calculated for a high-resolution reference solution and for several explicit and implicit subgrid models at coarser resolution. The reference solution shows that enstrophy and energy are removed from scales very close to the truncation scale and energy is transferred (backscattered) into the large scales. Some subgrid models are able to capture the removal of enstrophy from small scales, though none are scale-selective enough; however, none are able to capture accurately the energy backscatter. We propose a scheme that perturbs the vorticity field at each time step by the addition of a particular vorticity pattern derived by filtering the predicted vorticity field. Although originally conceived as a parametrization of energy backscatter, this scheme is best interpreted as an energy ‘fixer’ that attempts to repair the damage to the energy spectrum caused by numerical truncation error and an imperfect subgrid model. The proposed scheme improves the energy and enstrophy behaviour of the solution and, in most cases, slightly reduces the root mean square vorticity errors.

Estimates of small-scale non-orographic gravity-wave forcing in the mesosphere are investigated using Met Office middle atmospheric analyses. Such estimates are obtained using the ultrasimple spectral parametrization (USSP) gravity-wave scheme, currently employed operationally by the Met Office. A climatology of monthly zonal mean gravity-wave forcing from January 2005–December 2010 is presented, along with a discussion of estimated uncertainties and comparison with previous studies. Mesospheric gravity-wave forcing is found to be underestimated, consistent with the known seasonal evolution of extratropical mesospheric temperature biases within the Met Office assimilated dataset. The sensitivity of gravity-wave forcing to various parameters within the USSP scheme is investigated. Subsequent experiments diagnose the temperature response in a free-running version of the Unified Model when mesospheric gravity-wave forcing is increased through perturbing the energy scalefactor parameter within the USSP scheme and imposing momentum-flux conservation at the model lid. For physically justifiable perturbations to the USSP scheme, significant temperature responses of ∼10–25 K are seen for mesospheric polar regions under solstice conditions, highlighting the positive impact such changes could possibly have on known temperature biases within the Met Office assimilated dataset.

The Met Office weather and climate prediction model has recently upgraded its dynamical core—the model component that solves the equations of dynamics and thermodynamics on explicitly resolved scales. Designing, building, and testing the new dynamical core, known as ENDGame, took over 10 years, and was underpinned by a great deal of mathematical analysis. This chapter outlines some of the author’s mathematical contributions to the design of ENDGame, through his collaboration with the Met Office, and summarizes some of the positive impacts of the new dynamical core.

The accuracy of Met Office middle atmospheric analyses is investigated via direct validation against observational temperature data from two independent satellite instruments. A climatology of monthly zonal mean temperature biases from January 2005 to December 2010 is presented. For analyses produced before October 2009 there is a consistent cold bias of ∼30–40 K in the polar winter lower mesosphere, suggestive that gravity wave forcing in this region is too weak. Such cold biases are accompanied by smaller-magnitude cold biases in the opposing summer hemisphere, most likely associated with compensating warm biases from an insufficient gravity wave-driven circulation and cold biases resulting from deficiencies in radiative heating due to the operational ozone climatology. For analyses produced after October 2009, where the model lid was raised to include the upper mesosphere, cold biases in the winter lower mesosphere are reduced. However, systematic warm biases of ∼40–60 K in the summer upper mesosphere again suggest that gravity wave forcing within the operational system is underestimated. The absence of cold biases in the winter upper mesosphere also suggests that radiative errors contribute significantly in this region.

Accurate coupling between the resolved-scale dynamics and the parametrised physics is essential for accurate modelling of the atmosphere. Previous emphasis has been on the temporal aspects of this so-called physics–dynamics coupling problem, with little attention on the spatial aspects. When designing a model for numerical weather prediction there is a choice for how to vertically arrange the predicted variables, namely the Lorenz and Charney–Phillips grids, and there is ongoing debate as to which is the optimal. The Charney–Phillips grid is considered good for capturing the potential vorticity dynamics and wave propagation, whereas the Lorenz grid is more suitable for conservation. However the Lorenz grid supports a computational mode. It is argued here that the Lorenz grid is preferred for modelling the stably stratified boundary layer. This presents the question: which grid will produce more accurate results when coupling the large-scale dynamics to the stably stratified planetary boundary layer? The question is addressed by examining the ability of both the Lorenz and Charney–Phillips grids to capture the steady state of a set of equations that simultaneously represents both large-scale dynamics and the planetary boundary layer. The results show that the Charney–Phillips grid is able to capture accurately the steady boundary-layer solution provided the Richardson number is calculated without vertically averaging the shear. Averaging the shear suppresses the negative feedback of the shear on the diffusion coefficient; the positive feedback, via the vertical gradient of potential temperature, then leads to the formation of unrealistic step-like features. Copyright © 2012 Royal Meteorological Society

Currently, most operational forecasting models use latitude–longitude grids, whose convergence of meridians toward the poles limits parallel scaling. Quasi-uniform grids might avoid this limitation. Thuburn et al. and Ringler et al. have developed a method for arbitrarily structured, orthogonal C grids called 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. Some of the advantages and disadvantages of the hexagonal and triangular icosahedra, a “Voronoi-ized” cubed sphere, a Voronoi-ized skipped latitude–longitude grid, and a grid of kites in comparison to a full latitude–longitude grid are demonstrated. It is shown 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 that can be controlled using a diffusive advection scheme for potential vorticity.

The widely used shallow-atmosphere approximation is a geometrical approximation in which the metric departs from the usual Euclidean metric. This leads to a number of important consequences: shallow-atmosphere space is intrinsically curved (i.e. non-Euclidean), geodesics are not unique, the status of the centre of the Earth is uncertain, and position vectors are not well-defined. Vector semi-Lagrangian numerical models that use the shallow-atmosphere approximation must allow explicitly for the non-Euclidean geometry. During early testing of a new semi-implicit, semi-Lagrangian dynamical core, a semi-implicit (Crank–Nicolson) discretization of the vector departure point equation was found to lead to an instability in deep-atmosphere (i.e. Euclidean) geometry, but not in shallow-atmosphere geometry. The instability can be avoided by an alternative treatment in which the departure point equation is projected onto its horizontal and vertical components before discretization. Interestingly, this stable treatment of the deep-atmosphere case makes use of much of the mathematical machinery of the shallow-atmosphere departure point calculation. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

The spectral element method (SEM) has (with exact time integration) the desirable attribute of locally and globally conserving mass, energy and potential vorticity. It also scales well on massively parallel computers. Another desirable attribute of a numerical method for an atmospheric dynamical core is that it should have good numerical dispersion properties in order to accurately represent wave propagation and adjustment processes. Application of the SEM to the one-way wave equation is analysed to provide insight into its dispersion properties as a function of spectral order. For the lowest-order spectral truncation (linear) the SEM discretisation is formally equivalent to centred second-order finite differences on an Arakawa A grid. It consequently shares its poor dispersion properties, including energy propagation in the wrong direction for the short-wavelength half of the spectrum. Increasing the spectral truncation of the SEM to quadratic improves its dispersion properties for the long-wavelength part of the spectrum, but the problem of energy propagation in the wrong direction for the short-wavelength part remains. Further increasing the order of the spectral truncation not only fails to address the poor energy propagation at small scales, but also introduces new problems, including gaps in the spectrum of frequencies that can be represented, and localisation of eigenmode structures near element boundaries. Numerical integrations confirm that these SEM dispersion properties lead to reversed group velocities and to grid imprinting at spectral element boundaries. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

Atmospheric tracers are often observed to be functionally related, and these relations can be physically or chemically significant. It is therefore highly desirable that the transport schemes used in chemistry and chemistry-climate models should not disrupt such functional relations in unphysical ways through numerical mixing or, indeed, unmixing. Here, diagnostics are proposed that quantify numerical mixing by a transport scheme for a single tracer, two tracers that are nonlinearly related, and three (or more) tracers that add up to a constant. For the two-tracer test, the question of how physically reasonable the numerical mixing is can be addressed by using scatter/correlation plots. Truncation errors will, in general, result in scatter points deviating from the preexisting functional curve and thereby introduce numerical mixing between the tracers. The proposed diagnostics quantify the mixing in terms of the normalized distances between the pre-existing functional curve and scatter points, and divide it into three categories: real mixing and two types of spurious numerical unmixing. For the three-tracer test, we quantify, in terms of standard error norms, how nearly a transport scheme can preserve the sum by transporting the individual tracers. The mixing diagnostics do not require the knowledge of the analytical solution to the transport problem for the individual tracers. However, using an idealized flow field and spatial distributions facilitates the use of the mixing diagnostics in transport scheme development. Hence we propose to exercise the new mixing diagnostics using an idealized but highly deformational analytical flow field. Example results using the CSLAM (Conservative Semi-LAgrangian Multi-tracer) scheme are presented. Copyright © 2011 Royal Meteorological Society

A latitude–longitude grid is used by almost all operational atmospheric forecast models, and many research models. However, it is expected that the advantages of a latitude–longitude grid will become outweighed on massively parallel computers by data-communication bottlenecks. There is therefore renewed interest in quasi-uniform alternatives. This review surveys and assesses previously proposed horizontal grids for modelling the atmosphere over the sphere. Aspects of numerical accuracy likely to be affected by grid structure are discussed; particular attention is paid to computational modes and grid imprinting. Computational modes are potentially very serious, since they may be excited in realistic applications by boundary conditions, nonlinearity, physical forcing, and data assimilation. The geometry of polyhedra is reviewed due to its relation to numerical degrees of freedom, and hence to numerical wave dispersion and the possible existence of computational modes. All grids proposed to date have known problems or issues that merit further investigation. Orthogonal logically rectangular grids may be generated using conformal maps, but these suffer from singularities and resolution clustering. Resolution clustering may be avoided by using overset grids, but there are potential issues associated with the overlap regions. Alternatively, resolution clustering may be avoided, whilst retaining a logically rectangular grid, by giving up orthogonality; however, existing numerical schemes exploit orthogonality to obtain various properties thought to be important for accuracy, and it is not yet known whether these can also be obtained on non-orthogonal grids. Quasi-uniformity and orthogonality can be obtained without resolution clustering or overlaps by using non-quadrilateral grid cells, such as triangles, or pentagons and hexagons. However, when a staggered placement of variables is used to minimise dispersion errors for fast waves, non-quadrilateral grids support computational modes. In view of the lack of a single ideal grid, several topics meriting further investigation are identified. Copyright © 2011 Royal Meteorological Society and British Crown Copyright, the Met Office

Fil: Pulido, Manuel Arturo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Nordeste. Instituto de Modelado e Innovacion Tecnologica. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas Naturales y Agrimensura. Instituto de Modelado e Innovacion Tecnologica; Argentina. University of Toronto; Canada