Use of the Gibbs thermodynamic potential to express the equation of state in atmospheric models

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.The corrigendum to this article is in ORE: http://hdl.handle.net/10871/32395The 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.This work benefited from valuable discussions with Tommaso Benacchio, Geoffrey Vallis, Martin Willett, and Nigel Wood, as well as constructive reviews by Rupert Klein and an anonymous reviewer. It was funded in part by the Natural Environment Research Council under grant NE/N013123/1 as part of the ParaCon programme

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

Vertical discretizations giving optimal representation of normal modes: General equations of state

Previous work has identified a number of vertical discretizations of the nonhydrostatic compressible Euler equations that optimally capture the propagation of acoustic, inertio-gravity, and Rossby waves. Here, that previous work is extend to apply to a general equation of state, making it applicable to a wider range of geophysical fluid systems. It is also shown that several choices of prognostic thermodynamic variables and vertical staggering that were previously thought to be suboptimal can, in fact, give optimal wave propagation when discretized in an appropriate way. The key idea behind constructing these new optimal discretizations is to ensure that their corresponding linear system is equivalent to that of a certain, most fundamental, optimal configuration

Wave dispersion properties of compound finite elements

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.Mixed finite elements use different approximation spaces for different dependent variables. Certain classes of mixed finite elements, called compatible finite elements, have been shown to exhibit a number of desirable properties for a numerical weather prediction model. In two-dimensions the lowest order element of the Raviart-Thomas based mixed element is the finite element equivalent of the widely used C-grid staggering, which is known to possess good wave dispersion properties, at least for quadrilateral grids. It has recently been proposed that building compound elements from a number of triangular Raviart-Thomas sub-elements, such that both the primal and (implied) dual grid are constructed from the same sub-elements, would allow greater flexibility in the use of different advection schemes along with the ability to build arbitrary polygonal elements. Although the wave dispersion properties of the triangular sub-elements are well understood, those of the compound elements are unknown. It would be useful to know how they compare with the non- compound elements and what properties of the triangular sub-grid elements are inherited? Here a numerical dispersion analysis is presented for the linear shallow water equations in two dimensions discretised using the lowest order compound Raviart-Thomas finite elements on regular quadrilateral and hexagonal grids. It is found that, in comparison with the well known C-grid scheme, the compound elements exhibit a more isotropic dispersion relation, with a small over estimation of the frequency for short waves compared with the relatively large underestimation for the C-grid. On a quadrilateral grid the compound elements are found to differ from the non- compound Raviart-Thomas quadrilateral elements even for uniform elements, exhibiting the influence of the underlying sub-elements. This is shown to lead to small improvements in the accuracy of the dispersion relation: the compound quadrilateral element is slightly better for gravity waves but slightly worse for inertial waves than the standard lowest order Raviart-Thomas element.The work of John Thuburn was funded by the Natural Environment Research Council under the 'Gung Ho' project (grant NE/1021136/1)

A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes

Copyright © 2015 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. 290 (2015), DOI: 10.1016/j.jcp.2015.02.045A new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.Natural Environment Research Council under the “GungHo” projec

Numerical effects on vertical wave propagation in deep-atmosphere models

This is the final version of the article. Available from Wiley via the DOI in this record.Ray tracing techniques have been used to investigate numerical effects on the propagation of acoustic and gravity waves in a non-hydrostatic dynamical core discretised using an Arakawa C-grid horizontal staggering of variables and a Charney- Phillips vertical staggering of variables with a semi-implicit timestepping scheme. The space discretisation places limits on resolvable wavenumbers, and redirects the group velocity and the propagation of wave energy towards the vertical. The time discretisation slows the wave propagation while maintaining the group velocity direction. Wave amplitudes grow exponentially with height due to the decrease in the background density, which can cause instabilities in whole-atmosphere models. Although molecular viscosity effectively damps the exponential growth of waves above about 150 km, additional numerical damping might be needed to prevent instabilities in the lowermost thermosphere. These results are relevant to the Met Office Unified Model, and provide insight into how the stability of the model may be improved as the model’s upper boundary is raised into the thermosphere

Marginal stability of the convective boundary layer

This is the final version. Available from the American Meteorological Society via the DOI in this recordWe 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.Natural Environment Research Council (NERC

Vortex Erosion in a Shallow Water Model of the Polar Vortex

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.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.Robin Beaumont was supported during this research with a PhD studentship funded by an EPSRC Doctoral Training Grant