Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid

Quasi-uniform grids of the sphere have become popular recently since they avoid parallel scaling bottle- necks associated with the poles of latitude–longitude grids. However quasi-uniform grids of the sphere are often non- orthogonal. A version of the C-grid for arbitrary non- orthogonal grids is presented which gives some of the mimetic properties of the orthogonal C-grid. Exact energy conservation is sacrificed for improved accuracy and the re- sulting scheme numerically conserves energy and potential enstrophy well. The non-orthogonal nature means that the scheme can be used on a cubed sphere. The advantage of the cubed sphere is that it does not admit the computa- tional modes of the hexagonal or triangular C-grids. On var- ious shallow-water test cases, the non-orthogonal scheme on a cubed sphere has accuracy less than or equal to the orthog- onal scheme on an orthogonal hexagonal icosahedron. A new diamond grid is presented consisting of quasi- uniform quadrilaterals which is more nearly orthogonal than the equal-angle cubed sphere but with otherwise similar properties. It performs better than the cubed sphere in ev- ery way and should be used instead in codes which allow a flexible grid structure

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

Curl free pressure gradients over orography in a solution of the fully compressible Euler equations with implicit treatment of acoustic and gravity waves

Steep orography can cause noisy solutions and instability in models of the atmosphere. A new technique for modelling flow over orography is introduced which guarantees curl free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity. This mimetic property leads to better hydrostatic balance and better energy conservation on test cases using terrain following grids. Curl-free gradients are achieved by using the co-variant components of velocity over orography rather than the usual horizontal and vertical components. In addition, gravity and acoustic waves are treated implicitly without the need for mean and perturbation variables or a hydrostatic reference profile. This enables a straightforward description of the implicit treatment of gravity waves. Results are presented of a resting atmosphere over orography and the curl-free pressure gradient formulation is advantageous. Results of gravity waves over orography are insensitive to the placement of terrain-following layers. The model with implicit gravity waves is stable in strongly stratified conditions, with N∆t up to at least 10 (where N is the Brunt-V ̈ais ̈al ̈a frequency). A warm bubble rising over orography is simulated and the curl free pressure gradient formulation gives much more accurate results for this test case than a model without this mimetic property

Curl-Free Pressure Gradients over Orography in a Solution of the Fully Compressible Euler Equations with Implicit Treatment of Acoustic and Gravity Waves

Steep orography can cause noisy solutions and instability in models of the atmosphere. A new technique for modeling flow over orography is introduced that guarantees curl-free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity. This mimetic property leads to better hydrostatic balance and better energy conservation on test cases using terrain-following grids. Curl-free gradients are achieved by using the covariant components of velocity over orography rather than the usual horizontal and vertical components. In addition, gravity and acoustic waves are treated implicitly without the need for mean and perturbation variables or a hydrostatic reference profile. This enables a straightforward description of the implicit treatment of gravity waves. Results are presented of a resting atmosphere over orography and the curl-free pressure gradient formulation is advantageous. Results of gravity waves over orography are insensitive to the placement of terrain-following layers. The model with implicit gravity waves is stable in strongly stratified conditions, with NΔt up to at least 10 (where N is the Brunt–Väisälä frequency). A warm bubble rising over orography is simulated and the curl-free pressure gradient formulation gives much more accurate results for this test case than a model without this mimetic property

Numerical solution of the conditionally averaged equations for representing net mass flux due to convection

The representation of sub-grid scale convection is a weak aspect of weather and climate prediction models and the assumption that no net mass is transported by convection in parameterisations is increasingly unrealistic as models enter the grey zone, partially resolving convection. The solution of conditionally averaged equations of motion (multi-fluid equations) is proposed in order to avoid this assumption. Separate continuity, temperature and momentum equations are solved for inside and outside convective plumes which interact via mass transfer terms, drag and by a common pressure. This is not a convection scheme that can be used with an existing dynamical core -- this requires a whole new model. This paper presents stable numerical methods for solving the multi-fluid equations including large transfer terms between the environment and plume fluids. Without transfer terms the two fluids are not sufficiently coupled and solutions diverge. Two transfer terms are presented which couple the fluids together in order to stabilise the model: diffusion of mass between the fluids (similar to turbulent entrainment) and drag between the fluids. Transfer terms are also proposed to move buoyant air into the plume fluid and vice-versa as would be needed to represent initialisation and termination of sub-grid-scale convection. The transfer terms are limited (clipped in size) and solved implicitly in order to achieve bounded, stable solutions. Results are presented of a well resolved warm bubble with rising air being transferred to the plume fluid. For stability, equations are formulated in advective rather than flux form and solved using bounded finite volume methods. Discretisation choices are made to preserve boundedness and conservation of momentum and energy when mass is transferred between fluids. The formulation of transfer terms in order to represent sub-grid convection is the subject of future work

Multi-fluids for representing sub-grid-scale convection

Traditional parameterizations of convection are a large source of error in weather and climate prediction models and the assumptions behind them become worse as resolution increases. Multifluid modeling is a promising new method of representing sub- and near-grid-scale convection allowing for net mass transport by convection and non-equilibrium dynamics. The air is partitioned into two or more fluids which may represent, for example, updrafts and the nonupdraft environment. Each fluid has its own velocity, temperature and constituents with separate equations of motion. This paper presents two-fluid Boussinesq equations for representing sub-grid-scale dry convection with sinking and w = 0 air in fluid 0 and rising air in fluid 1. Two vertical slice test cases are developed to tune parameters and to evaluate the two-fluid equations: a buoyant rising bubble and radiative convective equilibrium. These are first simulated at high resolution with a single-fluid model and conditionally averaged based on the sign of the vertical velocity. The test cases are next simulated with the two-fluid model in one column. A model for entrainment and detrainment based on divergence leads to excellent representation of the convective area fraction. Previous multi-fluid modeling of convection has used the same pressure for both fluids. This is shown to be a bad approximation and a model for the pressure difference between the fluids based on divergence is presented

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests

Numerical methods for entrainment and detrainment in the multi-fluid Euler equations for convection

Convection schemes are a large source of error in global weather and climate models, and modern resolutions are often too fine to parameterise convection but are still too coarse to fully resolve it. Recently, numerical solutions of multi-fluid equations have been proposed for a more flexible and consistent treatment of sub-grid scale convection, including net mass transport by convection and non-equilibrium dynamics. The technique involves splitting the atmosphere into multiple fluids. For example, the atmosphere could be divided into buoyant updrafts and stable regions. The fluids interact through a common pressure, drag and mass transfers (entrainment and detrainment). Little is known about the numerical properties of mass transfer terms between the fluids. We derive mass transfer terms which relabel the fluids and derive numerical properties of the transfer schemes, including boundedness, momentum conservation and energy conservation on a co-located grid. Numerical simulations of the multi-fluid Euler equations using a C-grid are presented using stable and unstable treatments of the transfers on a well-resolved two-fluid dry convection test case. We find two schemes which are conservative, stable and bounded for large timesteps, and maintain their numerical properties on staggered grids

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

New limiter regions for multidimensional flows

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms. The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection