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Sylvie Gravel and Andrew Staniforth

Abstract

Within the context of a semi-Lagrangian shallow-water model, the dependence of forecast accuracy on the distribution of variable resolution and its robustness with respect to rapid variations in resolution is examined. This study also touches on the broader problem of designing a variable-resolution nested grid for regional modeling. It is demonstrated that the widely held belief that variable resolution induces severe noise problems at resolution interfaces—even for simple models—is not of universal applicability. In particular, no evidence of noise is found in the forecasts even when the resolution is changed abruptly by a factor of 3.5 across an internal boundary, thereby demonstrating the robustness of this particular variable-resolution technique. This result is achieved without any numerical smoothing technique other than that implicitly associated with the interpolation of a semi-Lagrangian scheme. The forecast produced on a uniform high-resolution mesh can be accurately reproduced for a limited time period on a subdomain at a fraction of the cost, by using a variable mesh where the resolution is gradually degraded away from this subdomain. The growth of the error variance when using such a mesh is an order of magnitude smaller than for one having the same number of degrees of freedom, except where the resolution changes abruptly at the boundary of the subdomain. It is concluded that variable resolution, using a smoothly varying mesh coupled with a semi-implicit, semi-Lagrangian integration scheme is an attractive approach to regional modeling.

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Jean Côté and Andrew Staniforth

Abstract

In Côté and Staniforth the efficiency of a semi-implicit spectral model of the shallow-water primitive equations was significantly improved by replacing the usual three-time-level Eulerian treatment of advection by a two-time-level semi-Lagrangian one. The Côté and Staniforth model nevertheless suffers from the important disadvantage of all global spectral models; viz. prohibitive expense at high resolution associated with the cost of the Legendre transforms.

A scheme is proposed that considerably improves the computational performance of the Côté and Staniforth scheme at high resolution. This is achieved by replacing the spectral discretization by a finite-element one, which is referred to as pseudo-staggering. This spatial discretization scheme uses an unstaggered grid yet doesn't propagate small-scale energy in the wrong direction, and no ad hoc measures are taken to avoid pole problems.

The proposed model was tested by comparing its forecasts with those of both the Côté and Staniforth model and an independent high-resolution Eulerian spectral control model. It was found that one can stably and accurately integrate the new model with time steps as long as three hours (which is approximately 18 times longer than the limiting time step of an Eulerian spectral model at equivalent resolution), without recourse to any divergence damping and with no evidence of any pole problem.

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Sylvie Gravel and Andrew Staniforth

Abstract

By generalizing the algorithm of Priestley for passively advected fields, a mass-conserving scheme for the coupled shallow-water equations is obtained. It is argued that the interpolation step of semi-Lagrangian schemes is the principal reason for their lack of formal conservation. The corrections introduced by the proposed algorithm to achieve conservation appropriately reflect the localized nature of the interpolation errors that induce its violation.

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John Thuburn and Andrew Staniforth

Abstract

Discretizations of the linearized shallow-water equations on a spherical C grid are considered. Constraints on the schemes' coefficients that ensure conservation of mass, angular momentum, and energy are derived. These results generalize previously published results to the case of nonuniform and rotated grids (but are restricted to the linearized equations). Grids with υ stored at the poles and grids with u and h stored at the poles are both considered. Energy conservation is shown to be problematic for grids with u and h at the poles.

It is also shown that an inappropriate averaging of the Coriolis terms leads to a misrepresentation of the Rossby modes with shortest meridional scale. The appropriate averaging is shown to be compatible with the constraints required for conservation, and, indeed, the energy-conserving averaging of the Coriolis terms improves the dispersion properties of Rossby modes.

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Elisabetta Cordero and Andrew Staniforth

Abstract

Three-time-level schemes for first-order differential equations are affected by the existence of computational modes. The Robert–Asselin filter was developed in an Eulerian context to selectively damp such modes while leaving the physical solution virtually untouched. Here previous analysis of this time filter for simple equation sets is extended to three-time-level semi-implicit semi-Lagrangian discretizations of the hydrostatic primitive equations. For Courant numbers greater than one-half, a surprising loss of selectivity is found when compared to that previously observed in the fully Eulerian context. Not only can the physical solution be undesirably more damped than the computational one, it can be significantly more damped. This problem is particularly serious in polar regions of a global gridpoint model due to the convergence of the meridians, although its severity might be reduced by using horizontal diffusion or spatial filtering to make the flow field more isotropic. (It is, however, no more serious there than elsewhere for a triangularly truncated spectral model because of its isotropic representation.) The problem's source is attributed to the current practice of integrating the equations along the trajectories but time filtering at grid points. Although time filtering along a trajectory would mitigate the problem, it would be costly in practice because of the additional computations required.

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Jean Côté and Andrew Staniforth

Abstract

Recently, it has been demonstrated that the semi-implicit semi-Lagrangian technique can be successfully coupled with a three-time-level spectral discretization of the barotropic shallow-water equations. This permits the use of time steps that are much larger than those permitted by the Courant-Friedrichs-Lewy (CFL) stability criterion for the corresponding Eulerian model, without loss of accuracy.

In this paper we show that it is possible to further quadruple the efficiency of semi-implicit semi-Lagrangian spectral models beyond that already demonstrated. A doubling of efficiency accrues from the use of the stable and accurate two-time-level scheme described herein. For semi-implicit semi-Lagrangian spectral models a further doubling of efficiency can be achieved by using a smaller computational Gaussian grid than the usual one, without incurring the significant loss of stability and accuracy that is observed for the corresponding Eulerian spectral model in analogous circumstances.

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Rodolfo Bermejo and Andrew Staniforth

Abstract

A method to convert conventional semi-Lagrangian schemes into quasi-monotone schemes is given. Numerical examples with linear and nonlinear transport equations demonstrate the ability of our modified semi-Lagrangian schemes to better maintain the shape of the solution in the presence of shocks and discontinuities.

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Andrew Staniforth and Nigel Wood

Abstract

Previous analysis of the hydrostatic primitive equations using a generalized vertical coordinate is extended to the deep-atmosphere nonhydrostatic Euler equations, and some special vertical coordinates of interest are noted. Energy and axial angular momentum budgets are also derived. This would facilitate the development of conserving finite-difference schemes for deep-atmosphere models. It is found that the implied principles of energy and axial angular momentum conservation depend on the form of the upper boundary. In particular, for a modeled atmosphere of finite extent, global energy conservation is only obtained for a rigid lid, fixed in space and time. To additionally conserve global axial angular momentum, the height of the lid cannot vary with longitude.

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Andrew Staniforth and Nigel Wood

Abstract

Normal-mode analyses are applied to various discrete forms of the one-dimensional, linearized, vertical acoustic equations in a height-based coordinate. First, the temporally discrete, spatially continuous equations are considered and the normal modes for a bounded system are compared to those of an unbounded system. Despite the use of a two-time-level discretization, a computational mode is found in the bounded case that is absent in the unbounded case. Second, the complete temporally and spatially discrete bounded system is considered and the normal modes and associated dispersion relation are derived. No computational modes are found. However, under certain limiting conditions, the temporal discretization leads to a distortion of the physical modes so that they resemble the computational mode of the spatially continuous bounded system. Implications for analyses based on spatially continuous equation sets are discussed.

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Andrew Staniforth and Jean Côté

Abstract

The semi-Lagrangian methodology is described for a hierarchy of applications (passive advection, forced advection, and coupled sets of equations) of increasing complexity, in one, two, and three dimensions. Attention is focused on its accuracy, stability, and efficiency properties. Recent developments in applying semi-Lagrangian methods to 2D and 3D atmospheric flows in both Cartesian and spherical geometries are then reviewed. Finally, the current status of development is summarized, followed by a short discussion of future perspectives.

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