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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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Clément Chouinard and Andrew Staniforth

Abstract

An algorithm is developed to derive hydrostatically balanced geopotentials at significant levels from radiosonde reports of significant-level temperatures and mandatory-level geopotentials and temperatures. It minimizes the square of the nonhydrostatic differences in a layer where at least one significant-level datum is reported and can be viewed as being a ID analysis step that returns an estimate of the departures from hydrostatic balance within the layer. The piecewise-polynomial interpolation of the minimization procedure is used to produce an expanded geopotential profile in any layer where significant-level data are reported, and the integrated minimization error can be used as a quality-control measure. The algorithm's performance has been evaluated using the global radiosonde dataset for a given synoptic time, and it is found that it produces equivalent layer-mean temperature errors that are generally smaller than radiosonde observational errors.

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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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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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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 Clive Temperton

Abstract

We present the firm application of the semi-implicit semi-Lagrangian integration technique to a finite-element barotropic model using the shallow-water equations on a variable-resolution. Two schemes based on this approach, differing only in their treatment of the rotational part of the wind field, are formulated and analyzed. A set of comparative experiments was performed using carefully balanced initial conditions (to eliminate spurious high-frequency oscillations); an Eulerian control integration was run at high resolution on a uniform grid. Both schemes are stable with timesteps at least six times longer than the limiting timestep of the corresponding Eulerian scheme using the same variable-resolution mesh. However, one scheme is consistently more accurate than the other. These results were explained by a theoretical analysis of the stability and accuracy of the schemes. We conclude that the semi-implicit semi-Lagrangian scheme is a promising technique for finite-element models as well as for finite-difference models.

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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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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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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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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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