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An Accurate and Efficient Finite-Element Global Model of the Shallow-Water Equations

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  • 1 Recherche en prévision numérique, Service de l'environnement atmosphérique, Dorval, P.Q., Canada
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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.

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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