Analysis of Numerically Induced Oscillations in 2D Finite‐Element Shallow‐Water Models Part I: Inertia‐Gravity Waves

Roux, Daniel Y. Le; Rostand, Virgile; Pouliot, B.

doi:10.1137/060650106

Cited by 72 publications

(104 citation statements)

References 36 publications

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“…In recent years, similar problems on triangular C-grids have been reported and investigated by other modelling groups (e.g. Le Roux et al, 2007;Danilov, 2010;Weller et al, 2012). Our analysis here provides more insight into the origin of the numerical noise from a different perspective.…”

Section: Discussionmentioning

confidence: 70%

“…On the triangular C-grids, spurious modes have also been noticed (e.g. Le Roux et al, 2007;Danilov, 2010;Weller et al, 2012). Some recent articles discussed this issue by analyzing the linearized shallow water equations and the representation of vector fields in a trivariate coordinate system (Thuburn, 2008;Danilov, 2010;Gassmann, 2011;Weller et al, 2012).…”

Section: Basic Operatorsmentioning

confidence: 99%

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The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids – Part 1: Formulation and performance of the baseline version

et al. 2013

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Abstract. As part of a broader effort to develop nextgeneration models for numerical weather prediction and climate applications, a hydrostatic atmospheric dynamical core is developed as an intermediate step to evaluate a finitedifference discretization of the primitive equations on spherical icosahedral grids. Based on the need for mass-conserving discretizations for multi-resolution modelling as well as scalability and efficiency on massively parallel computing architectures, the dynamical core is built on triangular C-grids using relatively small discretization stencils. This paper presents the formulation and performance of the baseline version of the new dynamical core, focusing on properties of the numerical solutions in the setting of globally uniform resolution. Theoretical analysis reveals that the discrete divergence operator defined on a single triangular cell using the Gauss theorem is only first-order accurate, and introduces grid-scale noise to the discrete model. The noise can be suppressed by fourth-order hyper-diffusion of the horizontal wind field using a time-step and grid-size-dependent diffusion coefficient, at the expense of stronger damping than in the reference spectral model.A series of idealized tests of different complexity are performed. In the deterministic baroclinic wave test, solutions from the new dynamical core show the expected sensitivity to horizontal resolution, and converge to the reference solution at R2B6 (35 km grid spacing). In a dry climate test, the dynamical core correctly reproduces key features of the meridional heat and momentum transport by baroclinic eddies. In the aqua-planet simulations at 140 km resolution, the new model is able to reproduce the same equatorial wave propagation characteristics as in the reference spectral model, including the sensitivity of such characteristics to the meridional sea surface temperature profile.These results suggest that the triangular-C discretization provides a reasonable basis for further development. The main issues that need to be addressed are the grid-scale noise from the divergence operator which requires strong damping, and a phase error of the baroclinic wave at medium and low resolutions.

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Section: Discussionmentioning

confidence: 70%

Section: Basic Operatorsmentioning

confidence: 99%

The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids – Part 1: Formulation and performance of the baseline version

et al. 2013

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“…Le Roux et al (1998) gave the first review of available choices. More recent mathematical and numerical analysis of finite element pairs for gravity and Rossby waves are provided in Le Roux et al (2007, , and . Hanert et al (2005) proposed to use the P NC 1 -P 1 pair, following Hua and Thomasset (1984).…”

Section: Introductionmentioning

confidence: 99%

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

et al. 2010

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We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hy- drostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic threedimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax-Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward.

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“…The properties mentioned above explain why a C-grid staggering seem to be a natural choice for the discretization of the equation of geophysical fluid dynamics on unstructured meshes such as triangular or even general polygonal grids [13]. Indeed, in a number of works it was shown that the triangular Cgrid discretization, and its finite-element counterpart, the RT 0 -P 0 discretization [12] ensures an accurate representation of propagating long surface waves (see, e. g., [10]) and allows the geostrophic balance to be maintained at a local level [14]. The RT 0 -P 0 discretization is more flexible and can be formulated on arbitrary triangular meshes, its relation to the standard triangular C-grid is discussed in [16].…”

Section: Introductionmentioning

confidence: 99%

Elementary dispersion analysis of some mimetic discretizations on triangular C-grids

Korn

Danilov

2017

Journal of Computational Physics

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Spurious modes supported by triangular C-grids limit their application for modelling large-scale atmospheric and oceanic flows. Their behavior can be modified within a mimetic approach that generalizes the scalar product underlying the triangular C-grid discretization. The mimetic approach provides a discrete continuity equation which operates on an averaged combination of normal edge velocities instead of normal edge velocities proper. An elementary analysis of the wave dispersion of the new discretization for Poincaré, Rossby and Kelvin waves shows that, although spurious Poincaré modes are preserved, their frequency tends to zero in the limit of small wavenumbers, which removes the divergence noise in this limit. However, the frequencies of spurious and physical modes become close on shorter scales indicating that spurious modes can be excited unless high-frequency short-scale motions are e↵ectively filtered in numerical codes. We argue that filtering by viscuous dissipation is more e cient in the mimetic approach than in the standard C-grid discretization. Lumping of mass matrices appearing with the velocity time derivative in the mimetic discretization only slightly reduces the accuracy of the wave dispersion and can be used in practice. Thus, the mimetic approach cures some di culties of the traditional triangular C-grid discretization but may still need appropriately tuned viscosity to filter small scales and high frequencies in solutions of full primitive equations when these are excited by nonlinear dynamics.

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Analysis of Numerically Induced Oscillations in 2D Finite‐Element Shallow‐Water Models Part I: Inertia‐Gravity Waves

Cited by 72 publications

References 36 publications

The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids – Part 1: Formulation and performance of the baseline version

The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids – Part 1: Formulation and performance of the baseline version

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

Elementary dispersion analysis of some mimetic discretizations on triangular C-grids

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