33 research outputs found
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
Using Power Diagrams to Build Optimal Unstructured Meshes for C-Grid Models
The Model for Prediction Across Scales (MPAS) for Ocean (-O), Sea-Ice (-SI) and Land-Ice (-LI), in addition to the Coastal Ocean Marine Prediction Across Scales (COMPAS) are two novel general circulation models designed to resolve coupled ocean-ice dynamics over variable spatial scales using non-uniform unstructured grids. Both models are based on a conservative mimetic finite-difference/volume formulation (TRiSK), in which staggered momentum, vorticity and mass-based degrees- of-freedom are distributed over an orthogonal 'primal-dual' mesh
Compatible finite element methods for numerical weather prediction
This article takes the form of a tutorial on the use of a particular class of
mixed finite element methods, which can be thought of as the finite element
extension of the C-grid staggered finite difference method. The class is often
referred to as compatible finite elements, mimetic finite elements, discrete
differential forms or finite element exterior calculus. We provide an
elementary introduction in the case of the one-dimensional wave equation,
before summarising recent results in applications to the rotating shallow water
equations on the sphere, before taking an outlook towards applications in
three-dimensional compressible dynamical cores.Comment: To appear in ECMWF Seminar proceedings 201
Numerical wave propagation for the triangular - finite element pair
Inertia-gravity mode and Rossby mode dispersion properties are examined for
discretisations of the linearized rotating shallow-water equations using the
- finite element pair on arbitrary triangulations in planar
geometry. A discrete Helmholtz decomposition of the functions in the velocity
space based on potentials taken from the pressure space is used to provide a
complete description of the numerical wave propagation for the discretised
equations. In the -plane case, this decomposition is used to obtain
decoupled equations for the geostrophic modes, the inertia-gravity modes, and
the inertial oscillations. As has been noticed previously, the geostrophic
modes are steady. The Helmholtz decomposition is used to show that the
resulting inertia-gravity wave equation is third-order accurate in space. In
general the \pdgp finite element pair is second-order accurate, so this leads
to very accurate wave propagation. It is further shown that the only spurious
modes supported by this discretisation are spurious inertial oscillations which
have frequency , and which do not propagate. The Helmholtz decomposition
also allows a simple derivation of the quasi-geostrophic limit of the
discretised - equations in the -plane case, resulting in a
Rossby wave equation which is also third-order accurate.Comment: Revised version prior to final journal submissio
Mixed finite elements for numerical weather prediction
We show how two-dimensional mixed finite element methods that satisfy the
conditions of finite element exterior calculus can be used for the horizontal
discretisation of dynamical cores for numerical weather prediction on
pseudo-uniform grids. This family of mixed finite element methods can be
thought of in the numerical weather prediction context as a generalisation of
the popular polygonal C-grid finite difference methods. There are a few major
advantages: the mixed finite element methods do not require an orthogonal grid,
and they allow a degree of flexibility that can be exploited to ensure an
appropriate ratio between the velocity and pressure degrees of freedom so as to
avoid spurious mode branches in the numerical dispersion relation. These
methods preserve several properties of the C-grid method when applied to linear
barotropic wave propagation, namely: a) energy conservation, b) mass
conservation, c) no spurious pressure modes, and d) steady geostrophic modes on
the -plane. We explain how these properties are preserved, and describe two
examples that can be used on pseudo-uniform grids: the recently-developed
modified RT0-Q0 element pair on quadrilaterals and the BDFM1-\pdg element pair
on triangles. All of these mixed finite element methods have an exact 2:1 ratio
of velocity degrees of freedom to pressure degrees of freedom. Finally we
illustrate the properties with some numerical examples.Comment: Revision after referee comment
Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
A mixed mimetic spectral element method is applied to solve the rotating
shallow water equations. The mixed method uses the recently developed spectral
element histopolation functions, which exactly satisfy the fundamental theorem
of calculus with respect to the standard Lagrange basis functions in one
dimension. These are used to construct tensor product solution spaces which
satisfy the generalized Stokes theorem, as well as the annihilation of the
gradient operator by the curl and the curl by the divergence. This allows for
the exact conservation of first order moments (mass, vorticity), as well as
quadratic moments (energy, potential enstrophy), subject to the truncation
error of the time stepping scheme. The continuity equation is solved in the
strong form, such that mass conservation holds point wise, while the momentum
equation is solved in the weak form such that vorticity is globally conserved.
While mass, vorticity and energy conservation hold for any quadrature rule,
potential enstrophy conservation is dependent on exact spatial integration. The
method possesses a weak form statement of geostrophic balance due to the
compatible nature of the solution spaces and arbitrarily high order spatial
error convergence