Staggering Of Variables Research Articles

Sea-ice dynamics on triangular grids

We present a discretization of the dynamics of sea-ice on triangular grids. Our numerical approach is based on the nonconforming Crouzeix-Raviart finite element. An advantage of this element is that it facilitates the coupling to an ocean model that employs an Arakawa C-type staggering of variables. We show that the Crouzeix-Raviart element implements a discretization of the viscous-plastic and elastic-viscous-plastic stress tensor that suffers from unacceptable small scale noise in the velocity field. To resolve this issue we introduce an edge-based stabilization of the Crouzeix-Raviart element. Through a blend of theoretical considerations, based on the Korn inequality, and numerical experiments we show that the stabilized Crouzeix-Raviart element provides a stable discretization of sea-ice dynamics on triangular grids that is relevant for sea-ice modelling in ocean and climate science.

A conservative discretization of the shallow-water equations on triangular grids

A structure-preserving discretization of the shallow-water equations on unstructured spherical grids is introduced. The unstructured grids that we consider have triangular cells with a C-type staggering of variables, where scalar variables are located at centres of grid cells and normal components of velocity are placed at cell boundaries. The staggering necessitates reconstructions and these reconstructions are build into the algorithm such that the resulting discrete equations obey a weighted weak form. This approach, combined with a mimetic discretization of the differential operators of the shallow-water equations, provides a conservative discretization that preserves important aspects of the mathematical structure of the continuous equations, most notably the simultaneous conservation of quadratic invariants such as energy and enstrophy. The structure-preserving nature of our discretization is confirmed through theoretical analysis and through numerical experiments on two different triangular grids, a symmetrized icosahedral grid of nearly uniform resolution and a non-uniform triangular grid whose resolution increases towards the poles.

Numerical effects on vertical wave propagation in deep‐atmosphere models

Ray‐tracing techniques have been used to investigate numerical effects on the propagation of acoustic and gravity waves in a non‐hydrostatic dynamical core discretized 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 discretization places limits on resolvable wavenumbers, and redirects the group velocity and the propagation of wave energy towards the vertical. The time discretization 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.

Numerical Effects on Wave Propagation in Atmospheric Models

AbstractRay tracing techniques have been used to investigate numerical effects on the propagation of acoustic waves in a non-hydrostatic dynamical core discretised using an Arakawa C-grid horizontal staggering of variables (Arakawa & Lamb 1977) and a Charney-Phillips vertical staggering of variables (Charney & Phillips 1953) with a semi-implicit timestepping scheme. It is found that the space discretisation places limits on resolvable wavenumbers and redirects the group velocity of waves towards the vertical. Wave amplitudes grow exponentially with height due to the decrease in the background density, which can cause instabilities in whole-atmosphere models. However, the inclusion of molecular viscosity and diffusion acts to damp the exponential growth of waves above about 150 km. This study aims to demonstrate the extent to which numerical wave propagation causes instabilities at high altitudes in atmosphere models, and how processes that damp the waves can improve these model’s stability.

Formulation of an unstructured grid model for global ocean dynamics

A conservative discretization of the ocean primitive equations for global ocean dynamics is formulated on an unstructured grid. The grid consists of triangular cells with a C-type staggering of variables, where scalar variables are located at centers of grid cells and normal components of velocity are placed at cell boundaries. Reconstructions, necessitated by the staggering of variables, are chosen from a set of admissible reconstructions such that the discrete equations obey a weighted weak form, which guarantees discrete conservation properties. At the same time a spurious mode, specific to the triangular C-grid, is controlled in a way that is compatible with conservation properties. The conservation properties of the discrete model are verified. Numerical simulations are presented, ranging from idealized geophysical flows to an ocean simulation at eddy-permitting resolution, in order to substantiate the statement that global ocean dynamics can be formulated on triangular C-grids.

An improved version of SLICE for conservative monotonic remapping on a C‐grid

AbstractThis Note presents some further developments to the Semi‐Lagrangian Inherently Conservative and Efficient (SLICE) scheme. The revised scheme, termed C‐SLICE, is designed to exploit the good dispersion properties of a C‐grid staggering of mass and momentum variables in atmospheric models. Furthermore, several simplifications are made to the SLICE scheme, leading to improved computational efficiency without loss of accuracy. The revised scheme is evaluated using some standard tests from the literature. © Crown Copyright 2009. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

A finite‐element scheme for the vertical discretization of the semi‐Lagrangian version of the ECMWF forecast model

AbstractA vertical finite‐element (FE) discretization designed for the European Centre for Medium‐Range Weather Forecasts (ECMWF) model with semi‐Lagrangian advection is described. Only non‐local operations are evaluated in FE representation, while products of variables are evaluated in physical space. With semi‐Lagrangian advection the only non‐local vertical operations to be evaluated are vertical integrals. An integral operator is derived based on the Galerkin method using B‐splines as basis functions with compact support. Two versions have been implemented, one using piecewise linear basis functions (hat functions) and the other using cubic B‐splines. No staggering of dependent variables is employed in physical space, making the method well suited for use with semi‐Lagrangian advection.The two versions of the FE scheme are compared to finite‐difference (FD) schemes in both the Lorenz and the Charney–Phillips staggering of the dependent variables for the linearized model. The FE schemes give more accurate results than the two FD schemes for the phase speeds of most of the linear gravity waves. Evidence is shown that the FE schemes suffer less from the computational mode than the FD scheme with Lorenz staggering, although temperature and geopotential are held at the same set of levels in the FE scheme too. As a result, the FE schemes reduce the level of vertical noise in forecasts with the full model. They also reduce by about 50% a persistent cold bias in the lower stratosphere present with the FD scheme in Lorenz staggering (i.e. the operational scheme at ECMWF before its replacement by the cubic version of the FE scheme described here) and improve the transport in the stratosphere. Copyright © 2004 Royal Meteorological Society

Conservation of Potential Vorticity on Lorenz Grids

Abstract The quasigeostrophic equations formulated using the Charney–Phillips vertical staggering of variables are well known to possess an analog of the form of conservation of potential vorticity. It is shown that a similar analog is enjoyed by the quasigeostrophic equations formulated using the modified Lorenz staggering of variables.

Null-space-free methods for the incompressible Navier-Stokes equations on non-staggered curvilinear grids

Abstract The discrete Poisson equation used to enforce incompressibility in the projection method of Chorin has a null space. The null space often manifests itself in producing solutions with checkerboard pressure fields. The staggering of variables by Harlow and Welch effectively eliminates the null space; however, when it is used in the context of curvilinear coordinates its consistent implementation is complicated because it requires the use of contravariant velocity components and variable coordinate base vectors. In this paper we present and analyze a null-space-free approximate projection method, which is based on cartesian vector components and non-staggered grids. The approximate projection method has been implemented in the computer code HEMO. Several examples of two-and three-dimensional flows using HEMO are presented.

Simple test calculations concerning finite element applications to numerical weather prediction

AbstractDifferent finite element schemes are investigated with respect to their application in numerical weather prediction. Different methods of staggering of variables are considered. The tests concern the accuracy of a Rossby wave prediction and the generation of noise in a geostrophic adjustment process. Theoretical results concerning the noise level of different schemes are confirmed by computations with a one‐dimensional model. Favourable results were obtained by hybrid schemes, using different Galerkin treatments for different terms of the dynamic equations.